Information Entropy--Geometric Unification and Windowed Generation of Cosmological Terms: From Relative Entropy Hessian to Effective Action, Poisson--Euler--Maclaurin Finite-Order Discipline, and Geometric Entropy Decomposition of Friedmann Equations

In o ma ion En opyGeome ic Unica ion and Windowed
Gene a ion o Cosmological Te ms:
F om Rela i e En opy Hessian o Eec i e Ac ion,
PoissonEule Maclau in Fini e-O de Discipline, and Geome ic
En opy Decomposi ion o F iedmann Equa ions
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Ve sion: 1.5
Abs ac
Wi hin a unied ope a o measu e unc ion amewo k, we es ablish an o ganic assem-
bly connec ing
mul i-o de esponses o ela i e en opy
,
mas e -scale calib a ions o
sca e ing spec a
,
windowed eadou Toepli z/Be ezin comp essions
, and
Nyquis 
PoissonEule Maclau in (NPE) ni e-o de discipline
o closed de i a ions o
geo-
me ic eec i e ac ion
and
cosmological e ms
. Fi s , unde Eguchi egula ized di e gence
and Ama i
α
-geome y, we p o e cons uc ion o he Fishe Rao me ic and dual connec ions;
second, unde ace-class/ ela i e ace-class pe u ba ion and ene gy-die en iable sca e ing
heo y assump ions, we p esen a heo em-le el s a emen o he mas e scale ini y
φ′(ω)
π=−ξ′(ω) = 1
2π Q(ω), Q =−iS†∂ωS
poin ing ou dis ibu ional sense co ec ions a h esholds/long- ange po en ials. Nex , se-
lec ing PaleyWiene / de B anges / Ha dy en i onmen s, employing
symme ic smoo h
alloca ion
(
bg=pb
h
,
h=g∗˜g
), we place
Kw,h =P Mw1/2Cg·C˜gMw1/2P
in o Scha en ace
class and p o ide
explici uppe bounds
. Subsequen ly, uni ying Fou ie con en ions and
dis inguishing
Poisson ze o-aliasing c i e ion
(
∆<2π/B
) om
Shannon no-aliasing
econs uc ion
(
∆< π/B
) in hei mul iplica i e cons an die ences; unde double-laye
ail con ol o nea band-limi ed, we p o ide EM emainde wi h
ζ(2m)
explici cons an s.
Using Toepli zFIO diagonal- ype wa e- on ela ion, we p o e
windowingcomp ession
con olu ion singula i y non-inc ease
(holds on
T∗X 0
away om ze o cu , wi h band-
limi ed/nea band-limi ed windowing as global inclusion). In a minimal compu able model o
linea ized g a i y, we p o ide
explici coecien s
om
ou h-o de esponse
o
cu a-
u e quad a ic in a ian s
, he eby ob aining he
scale in eg al law
o olume e ms
Λe (µ)−Λe (µ0) = Zµ
µ0
Ξ(ω)dln ω, [Ξ] = L−2,
and p o ide sucien condi ions o posi i i y/mono onici y o
Ξ
plus local non-mono onici y
bounda ies a esonances/ h esholds. The ac ion unies as
Se [g] = Zd4x√−ghR−2Λe (µ)
16πG +αR2+βRµν Rµν +···i,
1
whe e
α, β
a e dimensionless. Using h ee-dimensional
S3
hea ke nelcoun ing unc ion
cu a u e docking example, we close he spec algeome ic in e p e a ion o FRW cu a u e
e m, demons a ing ia one-dimensional
δ
po en ial and AB sca e ing he windowing mecha-
nism o single-peak sa u a ion/peak- amily quasi-loga i hmic accumula ion. Appendices p o-
ide comple e p oo s o all heo ems, cons an es ima es and dimensional ables, plus ep o-
ducible expe imen al/nume ical sc ip essen ials.
MSC
: 53Bxx; 83C05; 58J35; 46E22; 47B35; 42A38; 94A17; 81U40
Keywo ds
: In o ma ion geome y; Eguchi egula ized di e gence; Fishe Rao me ic; Ama i
α
-connec ion; B egman/Hessian; spec al shi unc ion; Bi manK ein; Wigne Smi h g oup
delay; Toepli z/Be ezin comp ession; Scha en ace class c i e ia; Poisson summa ion; Eule 
Maclau in emainde cons an s; wa e- on se and Toepli zFIO; hea ke nel/SeeleyDeWi ;
spec al ac ion; unning acuum; FRW geome ic en opy decomposi ion
1 In oduc ion & His o ical Con ex
In o ma ion geome y cha ac e izes s a is ical mani olds ia Hessian me ics and
α
-connec ions
gene a ed by di e gences; second-o de esponse o ela i e en opy yields Fishe Rao me ic, hi d-
o de esponse co esponds o Ama iChen so enso and
α
-connec ion. B egman di e gence
induces dual a (Hessian) s uc u e and Legend e dual coo dina es in exponen ial amilies. In
spec alsca e ing heo y, he Li shi zK ein ace o mula and Bi manK ein iden i y ela e spec-
al shi unc ion
ξ
wi h sca e ing de e minan ; F iedelLloyd and Wigne Smi h uni y phase
de i a i e, g oup delay, and densi y-o -s a es die ence unde he same calib a ion. Hea ke nel/Seeley
DeWi expansion and spec al ac ion p inciple p o ide s anda d ools o b idging geome ic
in a ian swindowed spec a. This pape closes hese elemen s unde
heo em-le el assump-
ions
in o a logical chain om  ela i e en opymas e scalewindowingNPEhea ke nelFRW.
2 Model & Assump ions
2.1 Fou ie Con en ion, Va iables, and Window Ke nel Gene al Decla a ion
Fix
b
(ω) = ZR
e−iωx (x)dx, (x) = 1
2πZR
eiωx b
(ω)dω.
Th oughou , we uni o mly use
equency
ω
o eco d ene gy a iables ( eade s may iew
E≡
ω
). Windows
wµ
ake smoo hed loga i hmic windows, sa is ying
wµ∈C∞
0∩L∞
wi h
supp wµ⊂
[µ0, µ]
,
µ>µ0>0
; specically one may ake
wµ(ω) = ψ(ω)
ω, ψ ∈C∞
0, ψ ≡1
on
[µ0, µ]◦,
smoo hly cu o a
ω=µ0, µ
(i co e ing in e al a ound
ω≈0
,  s ake
µ0>0
hen ake
limi ).
Gene al decla a ion
: Readou ke nel
h
de aul s o
Bochne posi i e deni e
(
b
h≥0
,
b
h∈L1
), hus admi ing
bg=pb
h∈L2
such ha
h=g∗˜g
.
2
2.2 In o ma ion Di e gence and Dual Fla ness
Regula ized di e gence
D(θ∥θ0)
wi h second/ hi d/ ou h-o de esponses
gij =∂i∂jD|θ0, Tijk =∂i∂j∂kD|θ0,Kijkl =∂i∂j∂k∂lD|θ0.
Deno e
K:= Kijij
as ull con ac ion o ou h-o de esponse enso (un ela ed o
Kw,h
).
Induce Fishe Rao and
α
-connec ion:
Γ(α)ijk = Γ(0)ijk +α
2Tijk
. B egman di e gence
Dψ
makes
g=∇2ψ
,
∇(±1)
a .
2.3 Mas e Scale, Sca e ingSpec al Shi , and Th eshold Clauses
Sel -adjoin pai
(H0, H)
sa ises ace-class o ela i e ace-class pe u ba ion, wa e ope a o s
comple e;
S(ω)
uni a y and weakly die en iable. Spec al shi
ξ(ω)
sa ises
de S(ω) = e−2πiξ(ω)
.
Deni ion 1
(To al Sca e ing Phase)
.
Le
φ(ω) := 1
2iLog de S(ω),
aking he b anch consis en wi h h eshold phase eno maliza ion and con inuous as
ω→+∞
.
Then
φ′(ω) = 1
2i S−1∂ωS=1
2 Q(ω), Q =−iS†∂ωS,
hence
φ′(ω)
π=−ξ′(ω) = 1
2π Q(ω),
holding in dis ibu ional sense on disc e e h eshold se
Σ
.
2.4 Toepli z/Be ezin Comp ession and Readou
Take PaleyWiene / de B anges / Ha dy space
H
, o hogonal p ojec ion
P
(
no m
|P|= 1
). Le
w∈C∞
0∩L∞
,
h=g∗˜g
as abo e.
Deni ion 2
(Rela i e Spec al P ojec ion Die ence)
.
Deno e
Π
as he dis ibu ional ke nel o
ela i e spec al p ojec ion die ence
o sel -adjoin pai
(H0, H)
in ene gy ep esen a ion
(equi alen o ela i e spec al measu e), such ha
(Kw,hΠ) = Zw(ω) [h∗!ρ el](ω)dω,
whe e
ρ el(ω) = φ′(ω)
π=1
2π Q(ω)
,
Q=−iS†∂ωS
.
Dene
Kw,h := P Mw1/2Cg·C˜gMw1/2P, Obs(w, h) = (Kw,hΠ) = Zw(ω) [h∗!ρ el](ω)dω.
3
2.5 NPE Discipline and Nea Band-Limi ed
S ic ly band-limi ed
:
supp b
⊂[−B, B]
.
Nea band-limi ed
:
R|ω|>B |b
|dω ≤ε
and
R|ω|>B |b
|2dω ≤ε2
.
C i e ion dis inc ion
(de ailed in Theo em 3): Poisson ze o-aliasing e m
∆<2π/B
; Shannon
no-aliasing econs uc ion
∆< π/B
.
2.6 Eec i e Ac ion and Dimensions
Take
c=ℏ= 1
. Ac ion w i en as
Se =Zd4x√−ghR−2Λe (µ)
16πG +αR2+βRµνRµν +···i,
in ou dimensions
α, β
dimensionless;
[Λe ] = L−2
. Dimensional able in Appendix J.
3 Main Resul s (Theo ems and Alignmen s)
Theo em 3
(Rela i e En opy Hessian and
α
-Connec ion)
.
Second-o de esponse o ela i e en-
opy yields Fishe Rao me ic, hi d-o de esponse ia
Γ(α)ijk = Γ(0)ijk +α
2Tijk
gene a es
α
-
connec ion; B egman di e gence induces dual a (Hessian) s uc u e.
Theo em 4
(Mas e Scale T ini y: Sucien Condi ions and Th eshold Co ec ions)
.
Unde as-
sump ions o 3,
de S(ω) = e−2πiξ(ω)
and
ξ′(ω) = −1
2π Q(ω)
holds almos e e ywhe e; a
ω∈Σ
o long- ange po en ials holds in dis ibu ional sense wi h eno malized phase.
Theo em 5
(Poisson Ze o-Aliasing C i e ion and Shannon Recons uc ion C i e ion)
.
Unde he
p esen Fou ie con en ion, i
supp b
⊂[−B, B]
, hen
X
n∈Z
(n∆) = 1
∆X
k∈Zb
2πk
∆
wi h
k= 0
aliasing e ms
s ic ly ze o
i and only i
∆<2π/B
;
Shannon no-aliasing
econs uc ion
equi es
∆< π/B
.
Theo em 6
(NPE: Eule Maclau in Explici Cons an s and Nea Band-Limi ed Tails)
.
I
∈
C2m[a, b]
and is
(B, ε)
-nea band-limi ed,
|Rm| ≤ 2ζ(2m)
(2π)2m(b−a) sup
[a,b]| (2m)|+O(ε),sup | (2m)| ≤ C B2m| |∞.
Theo em 7
(Toepli zFIO Pseudolocali y and Singula i y Non-Inc ease)
.
Le
w∈C∞
,
h∈ S
,
P
be Toepli zFIO wi h diagonal- ype wa e- on ela ion. Fo any dis ibu ion
u
and any open cone
domain
U⋐T∗X 0
away om ze o cu ,
WFP MwChP u∩U⊆WF(u)∩U.
Rema k
8
.
When ene gy-shell windowing (band-limi ed/nea band-limi ed) excludes low- equency
neighbo hood o
|ξ| ≈ 0
, we ob ain global inclusion
WF(PMwChPu)⊆WF(u),
and when
WF(u)=∅
, he abo e inclusion is s ic .
4
Theo em 9
(Fou h-O de Response
→
Cu a u e Quad a ic Te ms: Minimal Compu able Model
and Coecien s)
.
Unde linea iza ion
gµν =ηµν+hµν
in ha monic gauge, decompose by scala / ans e se
aceless (TT) and dene windowed spec al weigh s
Ns=Zd4k k4W(k)|As(k)σ(k)|2,N =Zd4k k4W(k)|A (k)hTT(k)|2,
whe e
W
is de e mined by
ρ el, w, h
. Then
Z√−gK=c1Z√−g R2+c2Z√−g RµνRµν +
( o al de i a i e)
,
wi h
c1=Ns
36 , c2=Ns
12 +N
4.
No maliza ion decla a ion
: The deni ion o
Ns,
has abso bed all
(2π)
ac o s and measu e
cons an s in he unied Fou ie con en ion o his sec ion; unde die en con en ions, escaling is
equi ed acco dingly.
Theo em 10
(Volume Te m Scale In eg al Law and Posi i i y/Mono onici y o
Ξ
)
.
A e in o ma-
ion ee ene gy windowing,
Λe (µ)−Λe (µ0) = Zµ
µ0
Ξ(ω)dln ω, Ξ(ω) = ⟨K, ρ el⟩wω,h,[Ξ] = L−2.
I
ρ el(ω)≥0
and induced wo-poin ke nel wi h
wω, h
a e non-nega i e/Bochne posi i e deni e,
hen
Ξ(ω)≥0
and mono onically non-dec easing in
ln µ
; i
ρ el
sign- a iable, only window-a e aged
sense quasi-mono onici y is ob ained, o change
Ξ
o quad a ic o m o ob ain s ic non-nega i i y.
Th esholds/ esonance clus e s can cause local non-mono onici y, bu when peak amilies a e nea -
uni o mly dense in
ln ω
wi h slowly a ying weigh s,
Ξ
is nea ly cons an o e wide in e als, ex-
hibi ing quasi-loga i hmic accumula ion.
Theo em 11
(FRW Cu a u e Te m Spec alGeome ic Docking)
.
Th ee-dimensional cons an -
cu a u e mani old hea ke nel asymp o ic
T e− ∆∼(4π )−3/2hVol +
6ZR+O( 2)i, ↓0,
o
S3(L)
has
R= 6/L2
,
Vol = 2π2L3
. Windowed coun ing unc ion sub-leading e m
∝RR∝
κVol
consis en wi h FRW's
−κ/a2
e m; window shape only al e s coecien s, does no b eak
homogenei y and iso opy.
4 P oo s
4.1 Theo em 1 (Rela i e En opy Hessian and
α
-Connec ion)
Follows om Eguchi's con as unc ional and Ama iChen so enso deni ion. Realized ia
B egman po en ial in exponen ial amilies o dual a ness.
5

4.2 Theo em 2 (Mas e Scale T ini y: Sucien Condi ions and Th eshold Co -
ec ions)
P oo in h ee s eps:
1.
Spec al shi unc ion deni ion
: Dened by Li shi zK ein ace o mula.
2.
Sca e ing de e minan ela ion
: Bi manK ein iden i y yields
de S=e−2πiξ
.
3.
De i a i e ela ion
: Die en iabili y o s a iona y sca e ing de i es
ξ′=−(2π)−1 Q
.
Th eshold/long- ange po en ial cases hold in dis ibu ional sense wi h phase eno maliza ion
co ec ion.
4.3 Theo em 3 (Poisson Ze o-Aliasing C i e ion and Shannon Recons uc ion
C i e ion)
By Poisson o mula and p esen Fou ie con en ion,
b
(2πk/∆) = 0
(
k= 0
) i and only i
∆<2π/B
.
Shannon no-aliasing econs uc ion equi es s ic e condi ion:
∆< π/B
.
4.4 Theo em 4 (NPE: Eule Maclau in Explici Cons an s and Nea Band-
Limi ed Tails)
Employ DLMF's EM emainde cons an s and Be ns ein- ype de i a i e bounds. Nea band-limi ed
ails en e
O(ε)
.
4.5 Theo em 5 (Toepli zFIO Pseudolocali y and Singula i y Non-Inc ease)
Hö mande pseudolocali y yields
WF(Mwu)⊆WF(u)
, while
Ch
is smoo hing. Toepli zFIO
diagonal- ype wa e- on ela ion implies: o any
U⋐T∗X 0
,
WF(PMwChPu)∩U⊆WF(u)∩U.
Unde band-limi ed/nea band-limi ed windowing, can ake
U
co e ing en i e
T∗X 0
, hus
WF(PMwChPu)⊆WF(u)
.
4.6 Theo em 6 (Fou h-O de Response
→
Cu a u e Quad a ic Te ms: Mini-
mal Compu able Model and Coecien s)
F om linea ized decomposi ion ob ain
Kijkl
con ibu ion; i s ull con ac ion
K
ma ches
R2, RµνRµν
wi h coecien s
c1=Ns/36, c2=Ns/12 + N /4
.
Scala mode
: Linea ized cu a u e
R(1) =−6□σ
, hus
R2= 36 k4σ2
,
R(1)
µν Rµν(1) = 12 k4σ2
.
TT mode
:
R(1) = 0
,
RµνRµν =1
4k4(hTT)2
.
Ma ching windowed ou h-o de ke nel weigh s yields coecien s
c1,2
.
4.7 Theo em 7 (Volume Te m Scale In eg al Law and Posi i i y/Mono onici y
o
Ξ
)
Low- equency clus e (Poisson's
k= 0
) domina es olume e m. When
ρ el(ω)≥0
and ke -
nel/window non-nega i e/Bochne posi i e deni e,
Ξ≥0
; i
ρ el
sign- a iable, equi es window
a e aging o change o quad a ic o m.
6
Taube ian con ol when peak amilies nea -uni o mly dense in
ln ω
ensu es quasi-loga i hmic
in e als.
4.8 Theo em 8 (FRW Cu a u e Te m Spec alGeome ic Docking)
Use
S3
spec um
λn=n(n+ 2)/L2
, mul iplici y
(n+ 1)2
and Taube ian heo em o eco e hea
ke nel sub-leading e m and dock wi h FRW cu a u e e m.
5 Model Applica ions
5.1 One-Dimensional
δ
Po en ial: Single-Peak Sa u a ion and Quasi-Loga i hmic
Accumula ion
Take
V(x) = λδ(x)
. Unde he p esen uni con en ion
exp essing in ene gy a iable
E≡ω
,
phase shi w i en as
δ(E) = δk(E)=−a c an λ
2k(E), k(E) = √E(
may ake
2m= 1),
hence
ela i e densi y o s a es
ρ el(E) = 1
π
dδ
dE =1
π
dδ
dk
dk
dE 
below iden i y
E
wi h
ω.
Subsequen ly employ analy ic in eg a ion o loga i hmic window wi h Lo en zian peak o demon-
s a e single-peak sa u a ion/peak- amily quasi-loga i hmic accumula ion, compa ible wi h abo e
o mula. Fo smoo h loga i hmic window
I(µ;µ0) = Zµ
µ0
Γ
(ω−ω0)2+ Γ2
dω
ω
has closed o m
I(µ;µ0) = Γ
ω2
0+ Γ2ln µ
µ0−Γ
2(ω2
0+ Γ2)ln (µ−ω0)2+ Γ2
(µ0−ω0)2+ Γ2
+ω0
ω2
0+ Γ2ha c an µ−ω0
Γ−a c an µ0−ω0
Γi,
wo classes o
ln µ
exac ly cancel,
single-peak sa u a ion
; when peak amilies nea -uni o mly
dense in
ln ω
wi h slowly a ying weigh s, quasi-loga i hmic accumula ion eme ges.
Rep oducible expe imen al essen ials (example pa ame e s)
:
λ= 1
;
µ0= 10−3
, scan
µ
o
103
; window wid h smoo hing pa ame e
σ= 0.05
; ke nel
h(ω) = e−ω2/2σ2
h
ake
σh= 0.1
.
5.2 AB Sca e ing: Windowing TopologySpec al Densi y Die ence
Ideal AB model phase shi ene gy-independen ,
Q= 0
; ni e- adius/sc eened models in oduce
ene gy dependence, windowed die ence o ms eec i e con ibu ion o cu a u e/ opological e ms,
non-analy ic poin s co espond o s eps/cusps in
Ξ
.
7
6 Enginee ing P oposals
1.
G oup delay measu emen chain
: Measu e mul i-po
S(ω)
and die en ia e phase o
ob ain
Q(ω)
, cons uc
Ξ(ω)
and
Λe (µ)
cu es, NPE cons an s p o ide e o bands.
2.
Toepli z/Be ezin nume ical spec ology
: Implemen
Kw,h
and moni o
|Kw,h|1
; semiclas-
sical egime app oxima e ace by symbol in eg a ion and assess emainde by EM cons an s.
3.
FRW cu a u e windowing e ica ion
: On
S3/H3/
h ee- o us compa e hea ke nel sub-
leading e m wi h windowed coun ing unc ion, e i y spec algeome ic docking o
−κ/a2
.
7 Discussion
Mas e scale ini y holds unde ace-class/ ela i e ace-class and die en iabili y assump ions;
long- ange po en ials and h esholds co ec ed in dis ibu ional sense. Symme ic smoo h alloca-
ion o Toepli z/Be ezin p o ides checkable ace-class uppe bounds; NPE discipline o ms ni e-
o de e o budge by
ζ(2m)
cons an s and equency-domain ail con ol; windowingcomp ession
con olu ion non-inc eases singula i y. Fou h-o de esponse o
R2, RµνRµν
coecien s e iable
in minimal model; posi i i ymono onici y condi ions o
Ξ
explici , peak- amily s a is ics suppo
quasi-loga i hmic in e als. Windowed in e p e a ion o FRW cu a u e e m closed ia
S3
exam-
ple. Ex ensions o open sys ems o non-uni a y
S
equi e dissipa i e sca e ing amewo k, whe e
Q
loses posi i i y-p ese a ion.
8 Conclusion
Comple ing heo em-le el closu e om
in o ma ion di e gencemas e scalewindowing
NPEhea ke nelFRW
:
(i) Mas e scale ini y holds unde heo em-le el assump ions;
(ii) Toepli z/Be ezin comp ession en e s ace class ia symme ic smoo h alloca ion wi h explici
uppe bounds (
|P|= 1
);
(iii) Poisson ze o-aliasing and Shannon econs uc ion c i e ia sepa a ed wi h consis en cons an s;
(i ) EM emainde has
ζ(2m)
cons an s, nea band-limi ed ails con ollable;
( ) Windowingcomp essioncon olu ion non-inc eases singula i y (s ic ly non-inc easing unde
ene gy-shell windowing);
( i) Fou h-o de esponse o cu a u e quad a ic e m coecien s
explici ly e iable
;
( ii) Volume e m obeys scale in eg al law, posi i i y and quasi-loga i hmic mechanism o
Ξ
clea ;
( iii) FRW cu a u e e m spec algeome ic docking comple e.
These esul s p o ide e iable echnical ounda ion o unied scheme o in o ma ion geome y
×
spec alsca e ing
×
cosmology.
8
Acknowledgemen s, Code A ailabili y
No p op ie a y code used; appendices con ain ep oducible expe imen al/nume ical sc ip essen ials
and pa ame e ables.
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9