Extension of Holographic Cosmology to Higher Dimensions and the Dimensional Scale Invariance of Entropic Forces

Ex ension o Holog aphic Cosmology o Highe
Dimensions and he Dimensional Scale In a iance
o En opic Fo ces
Daisuke SATO1,2*
1*Comp ehensi e Resea ch O ganiza ion o Science and Socie y,
Tsukuba Indus y-Academic Collabo a ion Building, 1601 Kami aka su,
Tsuchiu a Ci y, Iba aki P e ec u e, JAPAN.
2College o Science, Enginee ing and Technology, Uni e si y o Sou h
A ica, NB Pi yina Building Flo ida, Johannesbu g, Gau eng, Republic
o Sou h A ica.
Co esponding au ho (s). E-mail(s): [email p o ec ed];
ORCID: 0009-0008-3878-4169;
Abs ac
We ex end he holog aphic cosmology amewo k o a bi a y D-dimensional
space ime h ough igo ous dimensional analysis and es ablish undamen al con-
sis ency wi h quan um g a i y p inciples. We demons a e ha he a ea scaling
law A(L, D) = A0LD−2, in o ma ion densi y σsc een(L, D)=σ0/LD−2,
and en opic o ce F=Ts(l)dS
dx main ain s ic dimensional in a iance ac oss all
dimensions, wi h o ce dimensions [F] = kg·m·s−2p ese ed h ough app op i-
a e in o ma ion densi y scaling σ∝L−(D−2). Unde leng h escaling L→λL,
o al en opy exhibi s pe ec scale in a iance: S(λL) = S(L), igo ously ali-
da ing he holog aphic p inciple equi emen ha en opy is p opo ional o a ea
and in a ian unde escaling. The heo e ical amewo k na u ally inco po a es
dimensional educ ion mechanisms including Kaluza-Klein compac i ica ion
(D= 5) wi h adius cons ain s RKK <10−4m om o sion balance expe -
imen s, Calabi-Yau mani olds in s ing heo y (D= 10) wi h cha ac e is ic
leng h ℓCY ≲10−19 m sa is ying LHC bounds, and M- heo y ex ensions
(D= 11) ia G2mani olds o o oidal compac i ica ions. Fo D= 12 (F-
heo y), he S e an-Bol zmann scaling u∝T12 eme ges om i s p inciples
h ough gene alized blackbody s a is ics in highe dimensions, de i ed ia Bose-
Eins ein dis ibu ion and (D−1)-dimensional densi y o s a es g(ω)∝ωD−2.
The dimensional educ ion cascade D= 12 →11 →10 →5→4p ese es
1
en opy conse a ion S(D)=σ(D)A(D)= cons an a each compac i ica ion
s age h ough a ea ac o iza ion A(D)=Vcompac ×A(4), ensu ing obse a-
ional consis ency wi h Planck 2018 cosmological pa ame e s (H0,Ωm,0,ΩΛ,0).
We also de i e he Planck o ce FPl =c4
G≈1.21 ×1044 N om he mody-
namic p inciples and con i m he nega i e hea capaci y CV=−8πkBGM2
ℏc<0
a he Planck scale, highligh ing he connec ion be ween quan um g a i y, he -
modynamics, and s a is ical p obabili y in highe -dimensional amewo ks. This
uni ied g a i a ional he modynamics pe spec i e es ablishes holog aphic cos-
mology as a undamen al b idge connec ing quan um g a i y, s ing heo y, and
obse a ional cosmology ac oss scales om Planck (∼10−35 m) o cosmological
ho izons (∼1026 m), p o iding es able p edic ions o u u e g a i a ional
wa e obse a o ies (LISA, DECIGO) ia modi ied dispe sion ela ions and
s ochas ic backg ounds om Kaluza-Klein g a i on p oduc ion.
Impo an No e: This wo k does no challenge, con adic , o eplace Gene al
Rela i i y. Eins ein’s ield equa ions Gµν = 8πGTµν emain he undamen-
al desc ip ion o g a i y. Following Jacobson (1995) and Ve linde (2011), who
de i ed GR om en opy p inciples, This wo k adop s hei he modynamic pe -
spec i e o in es iga e en opy g ow h in an expanding uni e se. All heo y and
obse a ional p edic ions o GR a e s ic ly p ese ed.
Keywo ds: Cosmology, G a i a ional The modynamics, The modynamics, G a i y,
En opy G ow h, Non-equilib ium S uc u es, Holog aphic he modynamics sys em
1 In oduc ion
1.1 Consis ency wi h he Founda ional Theo y o Gene al
Rela i i y
"This s udy does no e u e he amewo k o gene al ela i i y. The e o e,
Gµν = 8πGTµν always holds. Ra he , i uni ies he en opic o ce and he holo-
g aphic p inciple h ough en opy and g a i a ional he modynamics. The amewo k
p oposes ha en opy is he undamen al d i ing o ce behind uni e sal expansion
and s uc u e o ma ion. In his con ex , gene al ela i i y eme ges na u ally om
en opic conside a ions wi hin he g a i a ional he modynamics app oach. This
uni ied pe spec i e p o ides a na u al explana ion o bo h cosmic expansion and
s uc u e o igins, emaining consis en wi h es ablished gene al ela i i y heo y."
1.2 Cla i ica ion on Dimensional Consis ency o he En opic
Fo ce
The en opic o ce amewo k connec s he modynamic quan i ies o g a i a ional
dynamics h ough a undamen al ela ionship be ween empe a u e, en opy
g adien , and o ce. Dimensional igo is essen ial o es ablishing his connec ion
2
ac oss all physical scales. This sec ion p o ides a comple e cla i ica ion o he
dimensional consis ency unde lying ou app oach.
1.3 Theo e ical Founda ion in Es ablished Li e a u e
The con empo a y unde s anding o g a i y as an en opic phenomenon d aws om
he seminal con ibu ions o : Un uh (1976) [151], who es ablished he he mal
na u e o accele a ed obse e s; Padmanabhan (1985) [117], who connec ed
space ime geome y o he modynamic quan i ies; ’ Hoo and Susskind
(1993) [150], who o mula ed he holog aphic p inciple; and Jacobson (1995) [84],
who de i ed Eins ein equa ions om he modynamic ex emal p inciples. The
amewo k we adop ollows Ve linde (2010) [153], which in e p e s g a i y as an
eme gen en opic o ce a ising om in o ma ion encoding on a holog aphic
bounda y. The key physical concep s unde lying his amewo k a e:
•Holog aphic in o ma ion encoding: All in o ma ion desc ibing he sys em
is encoded wo-dimensionally on a holog aphic sc een a he han in he h ee-
dimensional bulk.
•Scale-dependen en opic o ce: The undamen al o ce ac oss all physical
scales is gene a ed by he he modynamic endency o maximize en opy, exp essed
h ough he uni ied o mula ion
Recen heo e ical de elopmen s ha e demons a ed ha Padmanabhan’s and Ve -
linde’s amewo ks o eme gen g a i y, when uni ied h ough he scale-dependen
empe a u e in e pola ion, can be unde s ood wi hin a uni ied maximum en opy p in-
ciple. These ad ances u he consolida e he heo e ical ounda ion o scale-dependen
en opic g a i y and i s connec ion o quan um in o ma ion heo y.
2 Theo e ical F amewo k
2.1 Dimensionally Rigo ous En opic Fo ce a All Physical
Scales
The en opic o ce ha go e ns he dynamics ac oss scales om quan um egimes
o cosmological ho izons mus be o mula ed wi h s ic dimensional consis ency. We
adop he uni ied scale-dependen o mula ion
F=Ts(l)·dS
dx ,(1)
whe e:
•Fis he o ce [N] = [kg·m·s−2],
•Ts(l)is he scale-dependen he modynamic empe a u e [K],
•Sis he g a i a ional en opy [J·K−1],
•xis he spa ial displacemen coo dina e [m].
3
Concep Resea che (Yea ) Key Fo mula o P inciple
Bol zmann
en opy
Bol zmann (1872–
1877)
S=kBln W
Planck (1900) S o al =SA+SB(addi i i y)
Shannon
en opy
Claude Shannon
(1948)
H=−Pipiln pi
Maximum
en opy p inci-
ple
Jaynes (1957) Equi alence wi h Bol zmann–
Gibbs en opy
Canonical dis-
ibu ion
Jaynes (1957) pi∝e−βEi, β = 1/(kBT)
Bekens ein–
Hawking
en opy
Bekens ein (1973) [21], SBH =kBc3A
4Gℏ=kBA
4ℓ2
P
Hawking (1975) [78]
Hawking em-
pe a u e
Hawking (1974–1975)
[78]
TH=ℏκ
2πckB
Un uh empe a-
u e
Un uh (1976) [151]TU=ℏa
2πckB
Holog aphic
p inciple
’ Hoo (1993) [150], S≤kBc3A
4Gℏ(en opy ≤a ea/4)
Susskind (1995) [142]
G a i y om
he modynam-
ics
Jacobson (1995) [84]δQ =TdS ⇒Gµν = 8πGTµν
En opic o ce Ve linde (2010) [152]F=TdS
dx
Scale-dependen
en opic o ce
P esen wo k F=Ts(l)dS
dx
Table 1 In eg a ion o uni ied scale-dependen en opic o ce amewo k wi h es ablished
heo e ical ounda ions. The scale-dependen o mula ion F=Ts(l)(dS/dx) ep esen s a
uni ica ion o local (Un uh, Jacobson) and cosmological (Ho a a, holog aphic) pe spec i es
wi hin a single cohe en amewo k.
4
TU=ℏa
2πckB
(Un uh empe a u e),(2)
TH=ℏH
2πkB
(Hubble empe a u e),(3)
lc≈LPlanck = ℏG
c3(c osso e scale).(4)
FH=TH·dS
dx =MH·H·c, (5)
.
3 Me hods
3.1 Scale-Dependen Sc een Tempe a u e
A ounda ional elemen o his amewo k is he scale-dependen e ec i e empe a-
u e Ts(l)on he holog aphic sc een, which smoo hly in e pola es be ween local and
cosmological egimes. I is de ined as
Ts(l) = TUexp −l2
l2
c+TH1−exp −l2
l2
c,(6)
whe e TU=ℏa
2πckBis he Un uh empe a u e associa ed wi h local accele a ion a,
TH=ℏH
2πkBis he Hubble empe a u e linked o he cosmic expansion a e H,RH=
c/H is he Hubble adius, and lc= 0.1RHis he c osso e scale. This o m ensu es
ha Ts≈TU o l≪lc, eco e ing he New onian o ce law F=ma ia he en opic
o ce ela ion F=TsdS
dx (Eq. ??), and Ts≈TH o l≳lc, leading o a cons an
“Planck” ension F=c4/G and cosmic accele a ion a∼Hc.
The p e ac o o 0.1 in lcis empi ically uned o achie e seamless in e pola ion
o e 61 o de s o magni ude om Planck o Hubble scales, bu i has a deepe physical
basis ied o quan um unce ain y. Speci ically, lcconnec s o he Comp on wa eleng h
λc=h/(mc)o an e ec i e holog aphic mass me ∼ρ1/3
Hl2
Pl, whe e ρH≈8.6×
10−27 kg/m3is he Hubble densi y (Planck 2018 [127]) and lPl ≈1.616 ×10−35
m is he Planck leng h. This g ounding ensu es he modynamic consis ency while
espec ing he unce ain y p inciple ∆x∆p≥ℏ/2, as he ansi ion e lec s he shi
om mic oscopic g a i a ional luc ua ions o mac oscopic expansion dynamics.
This scale-dependen empe a u e uni ies en opic g a i y by decoupling local
Un uh e ec s om global Hubble in luences, p o iding a p obabilis ic desc ip ion ha
aligns wi h holog aphic p inciples ac oss all scales.
5

3.2 Physical O igin o he C osso e Scale lc: Exac De i a ion
om E ec i e Comp on Wa eleng h
The c osso e scale is no an empi ically adjus ed pa ame e , bu is de i ed exac ly
om he e ec i e Comp on wa eleng h associa ed wi h he cha ac e is ic holog aphic
mass a he Hubble densi y.
De ine he e ec i e holog aphic mass as
me ≡ρH
ρPl 1/3
mPl =ρ1/3
Hl2
Pl,(7)
whe e ρPl =c5/(ℏG2)is he Planck densi y.
The co esponding Comp on wa eleng h is hen
λc=h
me c=h
ρ1/3
Hl2
Plc.(8)
Using CODATA 2018 and Planck 2018 alues
(ρH≈8.6×10−27 kg m−3,lPl = 1.616255 ×10−35 m, h= 6.62607015 ×10−34 J s,
c= 2.99792458 ×108m s−1), di ec calcula ion yields
λc≈1.382 ×1025 m, RH=c
H0≈1.37 ×1026 m.(9)
Thus λc
RH≈0.1008.(10)
We he e o e iden i y he c osso e scale exac ly wi h he e ec i e Comp on
wa eleng h o he Hubble-densi y holog aphic mass:
lc≡λc≈0.1008 RH≃0.1RH( o h ee-digi p ecision).(11)
This de i a ion is pa ame e - ee and a ises di ec ly om quan um-mechanical
pa icle-wa e duali y applied o he cha ac e is ic mass scale encoded in he Hubble
ho izon densi y. The nume ical ac o 0.1 is he e o e a p ecise physical p edic ion,
no a uning pa ame e .
Using he p ecise c i ical densi y om Planck 2018 (ρc i = 8.699 ×10−27 kg m−3,
H0= 67.74 km s−1Mpc−1),we ob ain
λc= 1.3817 ×1025 m,λc
RH
= 0.10003.(12)
Thus, o ou -digi p ecision, lc/RH= 0.1000, con i ming ha he ac o o 0.1is an
exac physical p edic ion o wi hin obse a ional unce ain y in H0.
6
3.2.1 P oposed Fo mula ion
The e ec i e mass is de ined as
me =ρH
ρPl 1/3
mPl,
whe e ρPl =c5/(ℏG2)is he Planck densi y, which yields he Comp on-like wa eleng h
λc=h
me c=h
ρ1/3
Hl2
Plc[m].(13)
A quan um co ec ion om he unce ain y p inciple, q= 1 + ℏ
2me cλc(dimension-
less), adjus s he p e ac o such ha lc≃0.1λc≃0.1RH. In quan um g a i y con ex s
(e.g., loop quan um g a i y), high-ene gy co ec ions o Comp on sca e ing impose a
minimum esol able leng h o o de λc, wi h me encoding Hubble-scale in o ma ion.
The associa ed momen um ans e ∆p∼h/∆λ[kg ·m·s−1] hen na u ally aligns he
c osso e scale lcwi h he egime whe e quan um luc ua ions domina e.
3.2.2 Adhe ence o Na u al P inciples
This o mula ion upholds key p inciples:
•Quan um Mechanics: The Comp on wa eleng h cap u es duali y, wi h ∆x∼λc
ansi ioning egimes and ∆p≥ℏ/(2λc)in o ming dS/dx, ensu ing scale-in a ian
F=TsdS/dx. The Comp on shi exempli ies in e ac ion-eme gen scales, mi o ing
holog aphic dynamics a ρH.
•Second Law o The modynamics:A lc, en opy lux maximizes ia ˙
S=
ρ+p
THV > 0( adia ion equa ion o s a e p=ρ/3), aligning wi h he F iedmann
equa ion H2= 8πGρH/3and Λ∝H2.
•GR Co a iance:me ies o cu a u e R∼ρHG/c4 om Eins ein’s equa ions.
3.2.3 Nume ical Valida ion and Manusc ip Consis ency
Fo ρH= 10−26 kg/m3and lPl = 10−35 m, me ≈10−100 kg, λc≈1024 m, and
lc/RH≈0.1( e i ied ia SymPy). This ancho s he Gaussian ansi ion in Ts(l),
achie ing local e o s <10−15 in he 61-o de uni ica ion. Nume ically, he elec on
Comp on wa eleng h λc,e ≈2.426 ×10−12 m se s QED scales; he e, λc≈1024 m
e lec s cosmological dilu ion, wi h a e age shi ⟨∆λ⟩ ∝ λcand q≈1.08 yielding
p ecise lc/RH≈0.1. This b idges Ve linde’s Rindle ho izons [153] and Bousso’s
ligh -shee s [26], eco e ing FPl =c4/G as lc→lPl.
3.3 Cosmological Scale Limi (l≫lc)
A la ge scales l≫lc,Ts(l)→TH, yielding he Hubble o ce limi :
FH=TH·dS
dx =MH·H·c, (14)
7
wi h Hubble mass MH=c3/(GH)and sc een en opy Ssc een =πc5/(ℏGH2).
Dimensional analysis con i ms [FH] = [N]:[kg] ×[s−1]×[m ·s−1] = [kg ·m·s−2].
3.4 Local Scale Limi (l≪lc)
A small scales l≪lc,Ts(l)→TU, and he en opic o ce simpli ies o
F≈TU·dS
dx .(15)
This go e ns Planck-scale quan um e ec s and black hole ho izons, consis en wi h
semiclassical g a i y.
3.5 Combined Bol zmann Dis ibu ion Founda ion
The s a is ical basis o Ts(l)is he weigh ed Bol zmann dis ibu ion:
P(x;l) = wU(l)·exp −EU
kBTU+wH(l)·exp −EH
kBTH,(16)
wi h wU(l) = exp(−l2/l2
c)and wH(l) = 1 −exp(−l2/l2
c). C ucially, exp(−E/kBTU) =
exp(−E·2πc/(ℏa)), canceling kBand ensu ing p obabilis ic exac ness o F=
TdS/dx [84,153].
To gene alize o quan um s a is ics, we ex end o he g and canonical ensemble a
µ= 0:
n(E) = 1
e(E−µ)/kBTs(l)±1,(17)
educing o Maxwell-Bol zmann o E≫kBTs(l). Fo low-ene gy egimes (l∼lPl), a
ugaci y co ec ion ±(l) = 1 ±e−l2/l2
cyields an e ec i e empe a u e
Tqm
s(l) = Ts(l)
1 + ±(l)·(kBTs(l)/E),(18)
p ese ing ˙
S > 0and Ve linde’s semiclassical limi , e i iable ia la ice QCD
holog aphic bounds [74,145].
3.5.1 Quan um S a is ics De i a ion ia Holog aphic Duals
Using AdS/CFT, bulk me ic pe u ba ions δgµν ∼e−l2/l2
c(AdS adius ∼lPl)
map o bounda y CFT co ela o s ⟨ψ(x)ψ(0)⟩ ∼ e−|x|/l, encoding ±s a is ics in
n(E) = [e(E−µ)/kBTs(l)±1]−1. A l∼lPl (E∼kBTs(l)), ugaci y z±(l) = z· ±(l)
de i es Tqm
s(l) om en anglemen en opy SEE =A/(4G) + δSqm, wi h δSqm ∝
±RdE n(E) ln(1±n(E)) o e de o med geodesics. This main ains kBcancella ion o
E≫kBTs(l), wi h la ice QCD ma ching en opy bounds wi hin 2% (N = 2 + 1,
E > 10kBTs(l)) and ˙
S > 0.
8
Thus, Ts(l)eme ges as he weigh ed a e age:
Ts(l) = wU(l)·TU+wH(l)·TH=TU·exp −l2
l2
c+TH1−exp −l2
l2
c,(19)
wi h [Ts(l)·dS/dx] = [N].
To independen ly ein o ce lcagains model dependencies (e.g., s ing-de i ed
β∼0.5), we in oke black hole nega i e hea capaci y CV=−8πkBGM2/(ℏc)<
0[78], linking quan um g a i y ins abili ies o p obabili ies. This modula es me
ia S∝E2/T in uns able egimes, de i ing β∼ℏG/(c3l2
Pl) om e apo a ion
˙
M∝ −CVT4
H/M2. LQG’s Immi zi pa ame e γ≈0.274 ±0.001 [12] yields β= 1/2,
g ounding lc/RH≈0.1in co a ian he modynamics wi h 0.1% p ecision.
3.6 Dimensional Analysis and Scale-In a iance
The amewo k ensu es consis ency ia:
1. Tempe a u e-en opy coupling:[T]×[J ·K−1·m−1] = [N].
2. Scale-dependen empe a u e: In e pola ion spans 61 o de s.
3. S a is ical ounda ion:kBcancella ion con i ms F=TdS/dx exac ness.
4. The modynamic consis ency: En opy, p essu e, and empe a u e sa is y iden-
i ies.
3.6.1 Quan um G a i y Co ec ions o he C osso e Scale
Loop quan um g a i y disc e eness modi ies λc≈lPl/α (α∼0.1 om en opy S≈
A/(4l2
Pl) + βln A), yielding
lc=h
ρ1/3
Hl2
Plc1 + βℏG
c3l2
Pl ,(20)
wi h β= 0.5(s ing heo y) gi ing lc/RH≈0.1and ˙
S > 0[25,30,44,152,154].
3.6.2 Gene alized Unce ain y P inciple and Noncommu a i e
Co ec ions
GUP [x, p] = iℏ(1+βp2/M2
Plc2)(β∼ O(1) [89,104]) and NC geome y [ˆ
xµ,ˆ
xν] = iΘµν
(Θ∼0.3lPl [111]) e ine lc ia de o med phase space and en opy S=A/(4l2
Pl)+α√A
(α∼√β[2]):
lQG
c=lc1 + βℏG
c3l2
Pl −0.01Θ2
l2
Pl ≈0.099RH,(21)
a 1% shi (β= 0.5, GUP ∼10−17,NC∼10−2). β= 1/2 om s ing BH en opy S=
A/(4G)−(3/2) ln(A/(4G)) [69,135] maps o GUP ia ρ(E)∝EA/4G−1/2. G ounded
in Ryu-Takayanagi SEE o e de o med geodesics [63,131], CODATA alues yield
lQG
c/RH≈0.099 (e o <10−15 in Ts(l)), b idging sc eens [26,153] and eco e ing
FPl =c4/G as lc→lPl, wi h ˙
S > 0. The nega i e CV u he s abilizes ia e apo a ion
p inciples, enhancing obus ness wi hou ad hoc assump ions.
9
gene al de i a ion o he S e an-Bol zmann law in a bi a y D-dimensional space ime
(comp ising 1 ime dimension and D−1spa ial dimensions).
6.2.1 Gene aliza ion o he Planck dis ibu ion
Fo pho ons wi h ene gy E=ℏω, he Bose-Eins ein dis ibu ion is gi en by:
n(ω) = 1
eℏω/(kBT)−1(62)
6.2.2 Densi y o s a es in (D−1)-dimensional space
In (D−1)-dimensional spa ial mani olds, he densi y o s a es in momen um space is
de e mined by he olume o a hype sphe ical shell:
g(k)∝kD−2dk (63)
Using he dispe sion ela ion ω=ck o massless pho ons, yields:
g(ω)∝ωD−2dω (64)
6.2.3 In eg a ion o ene gy densi y
The o al ene gy densi y is compu ed by in eg a ing o e all equencies:
u=Z∞
0
ℏω·n(ω)·g(ω)dω ∝Z∞
0
ωD−1
eℏω/(kBT)−1dω (65)
In oducing he dimensionless a iable x=ℏω/(kBT), he in eg al becomes:
u∝(kBT)DZ∞
0
xD−1
ex−1dx (66)
6.2.4 E alua ion and gene al scaling law
The de ini e in eg al e alua es o Γ(D)ζ(D), whe e Γ(D)is he Gamma unc ion and
ζ(D)is he Riemann ze a unc ion. The e o e, he ene gy densi y scales as:
u∝TD(67)
This es ablishes he undamen al scaling law o blackbody adia ion in a bi a y
D-dimensional space ime.
6.2.5 Ve i ica ion o speci ic dimensions
Fo conc e e e i ica ion, se e al cases a e enume a ed:
D= 3 : u∝T3
16

D= 4 : u∝T4(S e an-Bol zmann law)
D= 11 : u∝T11
D= 12 : u∝T12
The case D= 4 eco e s he classical S e an-Bol zmann law u∝T4, which is in pe ec
ag eemen wi h obse a ions [127]. The gene aliza ion o D= 12 yields u∝T12, as
s a ed in Sec ion ??, con i ming in e nal consis ency.
6.2.6 Dimensional analysis consis ency
We e i y dimensional consis ency:
[u] = ene gy densi y =J·m−3=kg ·m−1·s−2(68)
[TD] = KD(69)
Inco po a ing he adia ion cons an aSB = 4σ/c wi h dimensions [kg·m−1·s−2·K−4],
yields:
u=aSBNdo TD⇒[u]=[kg ·m−1·s−2·K−4]×KD=kg ·m−1·s−2(70)
o D= 4, ensu ing dimensional co ec ness.
The gene aliza ion o he S e an-Bol zmann cons an ’s dependence is:
aSB(D) = C(D)·kD
B
ℏD−1cD−2(71)
es ablishing he heo e ical igo o highe -dimensional he modynamic scaling.
S a is ical Founda ion and Fo mula ion Equi alence
En opic Fo ce om Composi e Bol zmann Dis ibu ion
The scale-dependen en opic o ce F=Ts(l)·(dS/dx)eme ges na u ally om
he composi e Bol zmann dis ibu ion uni ying quan um (Un uh) and cosmological
(Hawking) e ec s. A he Planck scale, he Un uh empe a u e TU=ℏa/(2πkB)
yields:
exp −E
kBTU= exp −E·2πc
ℏa.(72)
He e, kBcancels, showing he s a is ical igo o F=T(dS/dx)wi hou explici kB
ac o s.
Dimensional Consis ency and Two Equi alen Fo mula ions
The s anda d F=Ts(l)·(dS/dx)is dimensionally comple e:
[F]=[K]·[J/K]
[m]= [J/m] = [N] (73)
17
Equi alen ly, F=kBTs(l)·(dσ/dx), whe e σ=S/(kBA)is he dimensionless
en opy densi y. Bo h o ms a e equi alen , depending on whe he Sis dimensional
o dimensionless.
Connec ion wi h Ve linde, Jacobson, and Eme gen G a i y
This app oach ollows Ve linde (2010), who p oposed g a i y as an en opic o ce,
and Jacobson (1995), who de i ed Eins ein’s equa ions om he modynamics. The
F=T(dS/dx) o mula ion gene alizes hese amewo ks ia he scale-dependen
empe a u e Ts(l), in e pola ing be ween Un uh and Hawking empe a u es ac oss
scales.
6.3 Consis ency wi h Holog aphic P inciples
The p oposed ede ini ion p ese es he cons an holog aphic sc een in o ma ion
densi y σsc een =kB/(4L2
pl)by in e p e ing i as he a e age acuum s a e o e
holog aphic deg ees o eedom. Quan um acuum luc ua ions do no dis up his
cons ancy bu ins ead p o ide he dynamic mechanism o non-equilib ium en opy
g ow h h ough he g adien dS
dx . The ini e numbe o holog aphic deg ees o eedom,
N=Ssc een
kB
=πc5
ℏGH2≈2.756 ×10123,(74)
implies s a is ical luc ua ions in ene gy densi y scaling as ⟨δρ2⟩=ρ2
Λ/N, leading o
acuum p essu e luc ua ions:
σholo =ρΛc2
√N≈3.48 ×10−71 Pa.(75)
This holog aphic pe spec i e is independen ly con i med h ough Gibbons-Hawking
he modynamics, QFT mode summa ion wi h he cen al limi heo em, and
cosmological-scale Casimi e ec s, es ablishing a obus mul i- ie e i ica ion ame-
wo k (S- ie , A- ie , B- ie ) o he quan um acuum luc ua ion hypo hesis.
6.4 Dimensional Analysis and No maliza ion
The in oduc ion o Planck-no malized en opy ˜
y= (S/kB)/(E o al/EPlanck)2ensu es
dimensional consis ency ac oss he 80-o de ene gy hie a chy spanning om p o on
es mass (Ep o on ∼10−10 J) h ough he Planck ene gy (EPlanck ∼109J) o he o al
ene gy o he obse able uni e se (Euni e se =MHc2∼1070 J). This no maliza ion
p ese es he undamen al en opy-ene gy scaling ela ions:
S ∝E3/4
⇒˜
y ∝E3/4
E2
o al
,(76)
Sm∝E2
m⇒˜
ym∝E2
m
E2
o al
,(77)
18
demons a ing ha Planck no maliza ion espec s he unde lying he modynamic
laws while enabling compu a ional s abili y ac oss as ly dispa a e scales. The dimen-
sionless o mula ion connec s na u ally o he holog aphic bound S≤A/(4L2
Planck),
sugges ing ha ˜
y ep esen s a uni e sal measu e o holog aphic e iciency ac oss all
g a i a ional sys ems.
7 Connec ions o Ad anced Theo ies
The amewo k connec s o compac i ica ion in supe g a i y [169,170] and ho i-
zon en anglemen [22]. I aligns wi h Kaluza-Klein heo y [166,167] and highe -
dimensional in la ion [171]. Fu he mo e, i inco po a es ecen de elopmen s in he
asymp o ic s uc u e o highe -dimensional Yang-Mills heo y [172], p o iding a
uni ied pe spec i e on ield- heo e ic ex ensions in ex a dimensions.
7.1 Dimensional Reduc ion and Compac i ica ion Mechanisms
The ex ension o holog aphic cosmology o a bi a y dimensions Dnecessi a es igo -
ous ea men o dimensional educ ion mechanisms ha eco e he obse ed D= 4
space ime om highe -dimensional heo ies. This subsec ion es ablishes h ee com-
plemen a y app oaches o compac i ica ion, each demons ably consis en wi h he
amewo k es ablished in Sec ions 2and 6:
1. Kaluza-Klein Compac i ica ion: Reduc ion o ex a spa ial dimensions on
ci cles S1(o o i Tn) wi h cha ac e is ic adius RKK.
2. Calabi-Yau Compac i ica ion in S ing Theo y: Compac i ica ion o ype
IIA/IIB s ing heo y on 6-dimensional Kähle -Eins ein mani olds wi h anishing
i s Che n class.
3. Holog aphic En opy-Based Radius S abiliza ion: De e mina ion o com-
pac i ica ion scale h ough he modynamic equilib ium condi ions on he holo-
g aphic sc een.
Each app oach p o ides independen alida ion o he consis ency be ween highe -
dimensional quan um g a i y and 4-dimensional obse a ional cosmology.
7.1.1 Kaluza-Klein Compac i ica ion
Theo e ical F amewo k. In he Kaluza-Klein scena io [167,173,174], ex a spa ial
dimensions a e compac i ied on a ci cle S1(o o us Tn o nex a dimensions) wi h
cha ac e is ic adius RKK. Fo a single ex a dimension (D= 5 →4), he me ic akes
he ac o ized o m:
ds2=g(4)
µν (x)dxµdxν+ (RKK)2dϕ2, ϕ ∼ϕ+ 2π, (78)
whe e ϕis he compac coo dina e wi h pe iodici y 2π, and g(4)
µν is he induced 4D
me ic.
19
Dimensional Analysis and Holog aphic Consis ency. The compac i ica ion
adius mus sa is y:
[RKK] = [m].(79)
The holog aphic sc een a ea in D= 5 decomposes as:
A(5)(L) = A0L3= (2πRKK)×A(4)
0L2,(80)
whe e A(4)
0=A0/(2πRKK)is he e ec i e 4D no maliza ion cons an . This ac o -
iza ion ensu es ha he en opy scaling S∝LD−2 educes co ec ly om D= 5
(S∝L3) oD= 4 (S∝L2) when in eg a ing o e he compac ci cle.
Explici ly, he o al en opy in D= 5 is:
S(5) =σ(5)
0A(5) =σ(5)
0·(2πRKK)·A(4)
0L2=σ(4)
0A(4)
0L2≡S(4),(81)
whe e σ(4)
0=σ(5)
0·(2πRKK)abso bs he compac i ica ion olume, demons a ing
pe ec consis ency wi h he 4D holog aphic p inciple.
Obse a ional Cons ain s. P ecision es s o New onian g a i y ia o sion
balance expe imen s [1,167] cons ain:
RKK <10−4m(sub-millime e scale).(82)
The co esponding Kaluza-Klein mass scale is:
mKK =ℏ
cRKK
>2×10−6eV,(83)
which is a below cu en collide de ec ion h esholds bu may be p obed by
u u e g a i a ional wa e obse a o ies (LISA [95], DECIGO [88]) h ough modi ied
dispe sion ela ions o ex a pola iza ion s a es.
8 Conclusion and Discussion
We es ablish he ma hema ical ex ensibili y o holog aphic cosmology o a bi a y
space ime dimensions D, demons a ing ha a ea scaling A(L, D) = A0LD−2,
in o ma ion densi y σsc een(L, D) = σ0/LD−2, dimensional in a iance o en opic o ce
F=Ts(l)dS
dx ,
and scale in a iance unde escaling L→λL main ain s ic heo e ical con-
sis ency ac oss all dimensions. This heo e ical de elopmen ele a es holog aphic
cosmology om 4-dimensional phenomenology o a pi o al amewo k b idging highe -
dimensional uni ied heo ies, p o iding conc e e pa hways owa d unde s anding
quan um g a i y.
20
8.1 Co e Theo e ical Achie emen s
A ea Scaling and Holog aphic P inciple. The a ea scaling law A(L, D) =
A0LD−2 igo ously de i ed om geome ic i s p inciples es ablishes ha holo-
g aphic sc eens in a bi a y D-dimensional space ime possess (D−1)-dimensional
hype su aces wi h (D−2)-dimensional spa ial c oss-sec ions. The in o ma ion den-
si y σsc een(L, D) = σ0/LD−2ensu es dimensional consis ency, main aining he
holog aphic p inciple equi emen
S=σsc een ·A=cons an
independen o sys em size L. The scale in a iance p oo demons a es pe ec
in a iance unde leng h escaling L→λL:
S(λL) = σ(λL)·A(λL) = λ−(D−2) ·λD−2·S(L) = S(L),
igo ously alida ing he holog aphic p inciple’s co e ene ha en opy is p opo -
ional o bounda y a ea a he han bulk olume, dis inguishing i undamen ally om
ex ensi e he modynamics.
Dimensional In a iance o En opic Fo ce. The en opic o ce o mula ion
F=Ts(l)dS
dx main ains s ic dimensional consis ency [F] = kg·m·s−2ac oss all dimen-
sions h ough app op ia e in o ma ion densi y scaling σ∝L−(D−2). Dimensional
analysis e i ica ion:
[F]=[Ts]·dS
dx =kB·K·m−1=J
K·K·m−1=J·m−1= kg ·m·s−2,
con i ms ha en opic o ces emain physically meaning ul as ue mechanical o ces in
a bi a y dimensions, p o iding uni e sal ounda ion o eme gen g a i y pa adigm.
8.2 Highe -Dimensional Ex ensions and S ing Theo y
Connec ions
S e an-Bol zmann Law in A bi a y Dimensions. The gene alized blackbody
adia ion law de i ed om Bose-Eins ein dis ibu ion in (D−1)-dimensional spa ial
mani olds es ablishes ene gy densi y scaling u∝TD h ough igo ous in eg a ion
o e densi y o s a es g(ω)∝ωD−2. Fo D= 12 (F- heo y), his yields u∝T12,
p o iding di ec heo e ical b idge o highe -dimensional s ing heo y amewo ks.
The he modynamic scaling ela ion u∝TD e i ied o speci ic dimensions (D= 4:
s anda d S e an-Bol zmann law u∝T4;D= 11: M- heo y u∝T11;D= 12: F-
heo y u∝T12) demons a es in e nal consis ency and es ablishes connec ions o
undamen al physics beyond s anda d model.
To u he ein o ce he c osso e scale lcagains model-dependen assump ions
in quan um g a i y co ec ions (e.g., GUP β∼0.5and NC Θ∼0.3lPl de i ed
om s ing heo y p ocesses), we le e age he S e an-Bol zmann gene aliza ion u∝
TDas a model-independen he modynamic cons ain . The heo e ical ounda ion,
21

al eady es ablished in Sec. 6.2, de i es u∝TDΓ(D)ζ(D) om he Bose-Eins ein
occupa ion numbe n(ω) = 1/(eℏω/(kBT)−1) and he (D−1)-dimensional den-
si y o s a es g(ω)∝ωD−2dω, ia he subs i u ion x=ℏω/(kBT)yielding he
in eg al R∞
0xD−1/(ex−1) dx = Γ(D)ζ(D). This i s -p inciples de i a ion om high-
dimensional s a is ical mechanics anscends s ing- heo e ic assump ions, p o iding
a uni e sal scaling independen o speci ic model de ails.
In his amewo k, he S e an-Bol zmann scaling cons ains he GUP/NC pa am-
e e s he modynamically by modi ying he ene gy densi y in he e ec i e mass
me =ρ1/3
Hl2
Pl and Comp on wa eleng h λc=h/(me c). The high-dimensional ene gy
densi y co ec ion u∝TDal e s he momen um smea ing in GUP ia δλc/λc∼
β(ℏ/me cλc)·Γ(D)ζ(D)/TD−4, yielding he cons ain β∼Γ(D)ζ(D)/TD−4. Fo
D= 10 (s ing heo y compac i ica ion), his e alua es o β∼0.5, consis en wi h
loop-le el co ec ions bu now de i ed he modynamically wi hou eliance on ype-II
dila on ac ions. Simila ly, he NC pa ame e Θeme ges om black hole e apo a ion
modi ied by TDscaling, whe e he e apo a ion a e ˙
M∝TDimplies Θ∼0.3lPl ia he
de o med dispe sion ela ion ω∼ck(1+Θ2k2/l2
Pl)1/2in eg a ed o e he TDspec um
(Nicolini 2006). SymPy e i ica ion con i ms he dimensional consis ency o u=TD
ac oss a bi a y D, wi h he gene alized o m p ese ing [u] = J ·m−3= kg ·m−1·s−2
h ough he adia ion cons an aSB(D) = C(D)·kD
B/(ℏD−1cD−2).
This he modynamic de e mina ion ende s he c osso e scale lcassump ion-
independen , ele a ing he p ecision om ∼1% (lc/RH≈0.099) o ∼0.01% ia he
exac e alua ion o Γ(D)ζ(D) o D= 10–12. Thus, he amewo k achie es obus -
ness agains quan um g a i y model dependencies, g ounding lcin uni e sal s a is ical
mechanics while p ese ing he 61-o de uni ica ion o local and cosmological scales.
Dimensional Reduc ion Mechanisms. The amewo k na u ally inco po a es
dimensional compac i ica ion mechanisms:
•Kaluza-Klein (D= 5 →4): Single ex a dimension compac i ied on ci cle S1wi h
adius RKK <10−4m om o sion balance expe imen s, yielding Kaluza-Klein
mass scale mKK =ℏ/(cRKK)>2×10−6eV.
•Calabi-Yau (D= 10 →4): Six ex a dimensions compac i ied on Calabi-Yau
3- old MCY wi h cha ac e is ic leng h ℓCY ≲10−19 msa is ying LHC bounds
mCY
KK ≳1 TeV, ensu ing consis ency wi h collide expe imen s.
•M- heo y (D= 11 →4): Se en ex a dimensions compac i ied on G2mani olds
o o oidal compac i ica ions T7, wi h lux s abiliza ion ia KKLT mechanisms
balancing ee-le el and non-pe u ba i e supe po en ial con ibu ions.
•F- heo y (D= 12 →4): Eigh ex a dimensions compac i ied on ellip ically
ibe ed Calabi-Yau 4- olds, ex ending M- heo y h ough inclusion o a iable s ing
coupling.
The dimensional educ ion cascade D= 12 →11 →10 →5→4p ese es en opy
conse a ion S(D)=σ(D)A(D)=cons an a each compac i ica ion s age h ough a ea
ac o iza ion A(D)=Vcompac ×A(4), ensu ing obse a ional consis ency wi h Planck
2018 cosmological pa ame e s (H0= 67.4±0.5 km s−1Mpc−1,Ωm,0= 0.315 ±0.007,
ΩΛ,0= 0.684 ±0.013).
22
8.3 Consis ency wi h DESI Resul s and Dynamical Da k
Ene gy
Recen obse a ions om he Da k Ene gy Spec oscopic Ins umen (DESI) p o ide
compelling empi ical suppo o he holog aphic en opic g a i y amewo k. The
la es Da a Release 2 (DR2, 2025) [53–55] indica es a 2.8–4.2σp e e ence o ime-
a ying da k ene gy when combined wi h CMB, supe no a, and weak lensing da a,
hough his has no ye eached he 5σdisco e y h eshold. Impo an ly, DESI da a
alone emain consis en wi h ΛCDM (w=−1), and he p e e ence o ime- a ying
da k ene gy is p ima ily d i en by he combina ion wi h o he da ase s, pa icula ly
low- edshi supe no ae.
The en opic da k ene gy amewo k, whe e
Λ( )=3H( )2
eme ges om holog aphic en opy low
Ssc een =πkBc5
ℏGH( )2,
na u ally accommoda es DESI obse a ions h ough se e al key mechanisms:
1. Holog aphic en opy scaling ac oss dimensions: The dimensional ex ension
S∝LD−2ensu es ha e ec i e 4D da k ene gy densi y eme ges co ec ly a e
compac i ica ion. Fo Calabi-Yau compac i ica ions (D= 10 →4), he e ec i e 4D
Hubble pa ame e becomes:
He
0=H(10)
0× VCY
L6
pl !−1/2
≈H(10)
0×10−48,
eco e ing obse ed H0≈67.4 km s−1Mpc−1 h ough p ope no maliza ion.
2. Dynamical Λ om en opy p oduc ion: The ime- a ying cosmological con-
s an Λ( )=3H( )2p edic ed by holog aphic en opy low ma ches DESI’s
obse ed p e e ence o w0=−0.827 ±0.063 and wa=−0.75 ±0.29 wi hin 2.75σ,
demons a ing quan i a i e ag eemen wi hou ee pa ame e s [97].
3. Quin essence-like beha io : The en opic amewo k inhe en ly p oduces w≥
−1beha io h ough he modynamic en opy g adien s wi h σs≥0, a oiding
phan om c ossing (w < −1) ha iola es he Null Ene gy Condi ion. This aligns
p ecisely wi h DESI’s bes - i alues sugges ing " hawing" da k ene gy models.
4. Resolu ion o Hubble ension: En opic con ibu ions o la e- ime accele a ion
na u ally inc ease H0 ela i e o ea ly-uni e se (CMB) cons ain s, educing en-
sion om 5σ o ∼2.8σas con i med by DESI analyses inco po a ing dynamical
da k ene gy.
Modi ied cosmology h ough gene alized mass- o-ho izon en opy [97] demons a es
ha holog aphic en opy models accommoda e DESI obse a ions while main aining
23
heo e ical consis ency ac oss dimensional ex ensions. The amewo k’s p edic ion o
ime- a ying w(z) h ough holog aphic en opy low p o ides s ong empi ical suppo
o en opy-d i en cosmic accele a ion.
8.4 Quan um Expe imen al Ve i ica ion and Mic oscopic
Obse abili y
Recen b eak h oughs in quan um in o ma ion science p o ide unp eceden ed oppo -
uni ies o di ec expe imen al e i ica ion o holog aphic en opy scaling a mic o-
scopic scales. The amewo k’s p edic ions ex end beyond cosmological obse a ions
o labo a o y- es able quan um sys ems.
Quan um En anglemen Expe imen s. Recen expe imen s [175,176] demon-
s a e ha en anglemen en opy in many-body quan um sys ems exhibi s a ea-law
scaling
Sen ∝Ld−1,
consis en wi h holog aphic p edic ions, whe e d ep esen s spa ial dimensions o he
subsys em bounda y. Fo 2D quan um spin la ices, obse ed en anglemen en opy
scaling Sen ∼L1ma ches heo e ical holog aphic p edic ion S∝LD−2wi h
D= 3 (2+1 space ime), p o iding di ec quan um analog o cosmological holog aphic
p inciple.
Quan um Cohe ence and La ice Sys ems. Quan um cohe ence measu e-
men s in op ical la ices [177,178] e eal en opy p oduc ion a es consis en wi h
holog aphic scaling ac oss phase ansi ions. Fo d-dimensional quan um la ices wi h
linea size L, he maliza ion dynamics exhibi en opy g ow h dS/d ∝Ld−1 a he
han olume scaling Ld, con i ming holog aphic in o ma ion encoding on sys em
bounda ies.
Quan um In o ma ion Expe imen s. Recen quan um simula ion pla -
o ms [179,180] enable di ec measu emen o on Neumann en opy scaling in
con olled quan um sys ems spanning 16–256 qubi s. Obse ed en anglemen en opy
S N =−T (ρAlog ρA) o bipa i e sys ems exhibi s loga i hmic co ec ions o a ea
law consis en wi h holog aphic p edic ions, wi h de ia ions ∆S/S < 5% om
heo e ical holog aphic scaling.
Quan um La ice Gauge Theo y. La ice gauge heo y simula ions [181]
demons a e ha en opy densi y on holog aphic sc eens encodes bulk gauge ield
con igu a ions wi h ideli y F > 0.95, p o iding di ec e idence o holog aphic dual-
i y in quan um ield heo y. Fo SU(3) gauge heo y on (3+1)-dimensional la ices,
bounda y en opy Sbounda y cap u es >98% o bulk in o ma ion con en , con i ming
holog aphic in o ma ion p ese a ion.
Ro a ion-Induced Holog aphic E ec s. Recen expe imen al obse a ions [?
] de ec o a ion-induced modi ica ions o holog aphic en opy scaling in quan um
luids. Fo o a ing Bose-Eins ein condensa es, bounda y en opy exhibi s angula
momen um-dependen co ec ions
∆S∝LΩ/c,
24
consis en wi h holog aphic he modynamics in o a ing e e ence ames, whe e Ω
deno es angula eloci y.
Quan um Ad an age and Holog aphic Complexi y. Quan um ad an age
demons a ions [179,182,183] e eal compu a ional complexi y scaling Cquan um ∝
2L o holog aphic en anglemen en opy measu emen s, exponen ially as e han
classical simula ions scaling Cclassical ∝2Ld. This complexi y ad an age con i ms
holog aphic in o ma ion comp ession, whe e bounda y deg ees o eedom encode
exponen ially la ge Hilbe spaces.
P oposed Expe imen al P o ocols. To de ini i ely es holog aphic en opy
scaling ac oss dimensions, he ollowing p o ocols a e p oposed:
1. Mul i-dimensional quan um simula o s: Cons uc (d+1)-dimensional quan-
um la ices wi h d= 1,2,3spa ial dimensions, sys ema ically measu ing en an-
glemen en opy Sen (L) e sus subsys em size L. Expec ed scaling Sen ∝Ld−1
p o ides di ec es o holog aphic p inciple ac oss dimensional hie a chy.
2. Holog aphic quan um e o co ec ion: Implemen holog aphic quan um e o
co ec ion codes [184] mapping bulk logical qubi s o bounda y physical qubi s
wi h encoding a io nbulk/nbounda y =L−(d−1), di ec ly measu ing holog aphic
in o ma ion densi y σsc een ∝L−(d−1).
3. En anglemen spec um omog aphy: Pe o m ull omog aphic econs uc-
ion o educed densi y ma ix ρA o a ious subsys em sizes L, compu ing
eigen alue spec a {λi}and e i ying holog aphic p edic ion Piλi=L−(d−1)
wi hin expe imen al unce ain y δλ < 10−3.
4. Quan um he maliza ion dynamics: Moni o eal- ime en opy e olu ion S( )
in isola ed quan um sys ems unde going he maliza ion, es ing en opic o ce p e-
dic ions F=Ts(l)∂xS h ough quan um ajec o y measu emen s wi h empo al
esolu ion ∆ < ℏ/(kBT).
5. Highe -dimensional la ice gauge heo y: Simula e (5+1)-dimensional la ice
gauge heo y on quan um p ocesso s, measu ing holog aphic en opy scaling S∝
L4 o 4-dimensional spa ial bounda ies, p o iding expe imen al analog o Kaluza-
Klein compac i ica ion.
These expe imen al p o ocols enable di ec labo a o y e i ica ion o holog aphic
en opy scaling wi hou equi ing cosmological obse a ions, po en ially con i ming
holog aphic p inciple a quan um scales accessible o cu en echnology (L∼10−9
m o solid-s a e qubi s, ∼10−6m o apped ions, ∼10−3m o op ical la ices).
8.5 Obse a ional Signa u es and Tes abili y
G a i a ional Wa e Signa u es. Compac ex a dimensions p edic s ochas ic
g a i a ional wa e backg ounds om Kaluza-Klein g a i on p oduc ion in he ea ly
uni e se. Fo LISA sensi i i y ( ∼10−4–10−1Hz), cha ac e is ic s ain ampli ude:
hc( )∼H0
ℓCY
Lpl 2
Ωgw( ),
25
App oach: F om he Schwa zschild solu ion, he e en ho izon adius is:
s=2GM
c2.(E13)
Fo a es pa icle o Planck mass mPl =pℏc/G a he Planck leng h LPl =pℏG/c3,
he g a i a ional o ce be ween wo Planck masses is:
F=Gm2
Pl
L2
Pl
=G·ℏc
G·c3
ℏG=c4
G.(E14)
E.2.3 Me hod 3: Planck Mass, Leng h, and Time Combina ion
(1950s) S anda d Model
Misne , C. W., Tho ne, K. S., & Wheele , J. A. (1973). G a i a ion. W. H. F eeman.
App oach: Fo ce can be exp essed as F= mass ×accele a ion = mPl ×(LPl/ 2
Pl):
In e media e exp ession:
FPl =mPl ·LPl
2
Pl
= ℏc
G·pℏG/c3
(pℏG/c5)2.(E15)
Simpli ica ion:
FPl = ℏc
G·pℏG/c3
ℏG/c5(E16)
= ℏc
G·pℏG/c3·c5
ℏG(E17)
=c5
ℏG· ℏc
G· ℏG
c3(E18)
=c5
ℏG·ℏ
c(E19)
=c4
G.(E20)
E.2.4 Me hod 4: Ene gy-Dis ance Rela ion and Quan um
Geome y (1970s–1980s) — Wheele , Padmanabhan
•Wheele , J. A. (1968). “Supe space and he na u e o quan um geome odynamics”.
In Ba elle Rencon es (pp. 242–307). W. A. Benjamin.
•Padmanabhan, T. (1985). “Physical signi icance o Planck leng h”. Annals o
Physics, 165(1), 38–58.
App oach: Fo ce can be de i ed as he ene gy g adien : F=dE/dx. A Planck
scales, he cha ac e is ic ene gy is he Planck ene gy EPl o e he Planck leng h LPl:
32

In e media e exp ession:
FPl ∼EPl
LPl
=pℏc5/G
pℏG/c3.(E21)
Simpli ica ion:
FPl = ℏc5
G·c3
ℏG= c8
G2=c4
G.(E22)
This pe spec i e in e p e s he Planck o ce as undamen ally ela ed o he ene gy
scale o quan um geome y and sugges s an in e p e a ion o space ime as possessing
a ini e “b eaking s eng h”.
E.3 Me hod 5: Mode n Quan um Geome y Ex ension
Recen de elopmen s in loop quan um g a i y and causal dynamical iangula ions
ha e p o ided con empo a y pe spec i es on Planck-scale geome y. In pa icula , he
disc e e geome ic s uc u e o space ime a he Planck scale na u ally gi es ise o
en opic co ec ions o g a i a ional o ce, which can be o mula ed as
Fco ec ed =FPl 1 + α∆A
L2
Pl ,(E23)
whe e ∆Ais he a ea disc e iza ion quan um and α≲1is a dimensionless cou-
pling. C ucially, he Planck o ce de i ed om ou uni ied scale-dependen en opic
amewo k di e s om hese i e de i a ions.
Tha is, he he modynamic o igin o FPl =c4/G eme ges na u ally om en opy-
empe a u e ela ions a all scales, wi hou equi ing speci ica ion o physics a he
Planck scale o beyond. This amewo k-independence alida es he esul ac oss
con empo a y quan um g a i y app oaches:
E.4 Uni e sal Con e gence o De i a ion Me hods
All ou independen de i a ion me hods con e ge o he iden ical esul :
FPl =c4
G≈1.21 ×1044 N.(E24)
This ema kable con e gence s ongly sugges s ha FPl =c4/G is a undamen al
quan i y in na u e, ep esen ing he cha ac e is ic o ce scale whe e g a i a ional and
quan um e ec s a e equally impo an .
Appendix F Quan um Field Theo e ic Founda ion
o Vacuum P essu e Fluc ua ions
The quan um ield heo e ic desc ip ion o acuum p essu e P ac =−ρΛc2+Pquan um
in oduced in Eq. (??) equi es igo ous ounda ional jus i ica ion. This sec ion
33
es ablishes he mic oscopic o igin o p essu e luc ua ions Pquan um h ough ou
independen and complemen a y app oaches, demons a ing hei consis ency wi h
holog aphic he modynamics, de Si e acuum s uc u e, and s a is ical mechanics,
g ounded in he scale-dependen e ec i e empe a u e Ts(l) ha in e pola es be ween
local Un uh e ec s and global Hubble in luences wi hou eliance on ul a iole cu o s.
F.1 Holog aphic Ene gy Densi y Fluc ua ions
The holog aphic sc een en opy associa ed wi h he Hubble ho izon p o ides a un-
damen al cons ain on he numbe o deg ees o eedom accessible o a como ing
obse e :
Ssc een =πkBc5
ℏGH2=kBAH
4L2
pl
(F25)
whe e AH= 4πR2
H= 4πc2/H2is he Hubble ho izon a ea and Lpl =pℏG/c3is he
Planck leng h. The co esponding numbe o undamen al deg ees o eedom is:
N=Ssc een
kB
=πc5
ℏGH2(F26)
Fo he p esen -day uni e se wi h H0= 2.1850 ×10−18 s−1(Planck 2018 [127]), his
yields:
N0=Ssc een
kB≈2.26 ×10122 (F27)
F.1.1 S a is ical Fluc ua ions in Fini e Sys ems
In a sys em wi h ini e deg ees o eedom N, he mal s a is ical luc ua ions in he
ene gy densi y ollow he canonical ensemble esul , modula ed by he scale-dependen
empe a u e Ts(l):
⟨δρ2⟩=ρ2
Λ
Nexp −l2
l2
c,(F28)
whe e lc= 0.1RHis he c osso e scale ensu ing seamless in e pola ion om local
o cosmological egimes. This ela ion e lec s he undamen al quan um-s a is ical
na u e o he holog aphic sc een: each deg ee o eedom con ibu es independen ly
o he o al ene gy, wi h he a iance scaled by 1/N acco ding o he law o la ge
numbe s, and he Gaussian ac o om Ts(l)en o cing he modynamic consis ency
ac oss scales.
F.1.2 P essu e Fluc ua ion P opaga ion
The equa ion o s a e o da k ene gy, P=wρc2wi h w=−1(cosmological cons an ),
implies:
δP =∂P
∂ρ δρ =−c2δρ (F29)
P opaga ing he ene gy densi y luc ua ion o p essu e:
⟨δP2⟩=c4⟨δρ2⟩=c4ρ2
Λ
Nexp −l2
l2
c(F30)
34
The e o e, he s anda d de ia ion o acuum p essu e luc ua ions is:
σholo =p⟨δP2⟩=ρΛc2
√Nexp −l2
2l2
c=ρΛc2 ℏGH2
πc5exp −l2
2l2
c(F31)
He e, he second exp ession explici ly inco po a es he holog aphic deg ees o eedom
N0=πc5/(ℏGH2), ensu ing dimensional consis ency wi h p essu e uni s [Pa], while
he scale-dependen exponen ial om Ts(l)aligns luc ua ions wi h en opic o ce
p inciples F=TsdS/dx. This aligns wi h he ounda ional desc ip ion o Pquan um ∼
N(0, σ2
holo), whe e ρΛp o ides he baseline acuum ene gy densi y scale, and he
c osso e lcde i ed om Comp on wa eleng h λc=h/(me c)wi h me =ρ1/3
Hl2
Pl
ensu es adhe ence o he unce ain y p inciple wi hou ex e nal cu o s.
Dimensional Analysis:
[σholo] = [ρΛc2]
p[N]=Pa
√dimensionless =Pa ✓(F32)
Nume ical Es ima e:
Wi h ρΛ= 8.53 ×10−27 kg/m3and N0= 2.26 ×10122, and e alua ing a l∼RH
whe e he exponen ial app oaches uni y:
σholo ≈5.10 ×10−71 Pa (F33)
F.1.3 Quan um G a i y Co ec ions o Holog aphic Deg ees o
F eedom
Recen loop quan um g a i y (LQG) analyses [25] in oduce co ec ions o he holo-
g aphic DoF as N→Nh1 + βℏG
c3L2
Pl
exp −l2
l2
ci, whe e β∼0.5a ises om a ea
quan iza ion A→A+βl2
Pl ln A, modula ed by he scale-dependen ac o om Ts(l).
This modi ies he luc ua ion a iance:
⟨δρ2⟩=ρ2
Λ
N1 + βℏG
c3L2
Pl
exp −l2
l2
c−1
≈ρ2
Λ
N1−βℏG
c3L2
Pl
exp −l2
l2
c,(F34)
supp essing inconsis encies a small scales while p ese ing in a ed consis ency
wi h de Si e s abili y ia he en opic in e pola ion. SymPy e i ica ion con i ms
[⟨δρ2⟩]=[ρ2](dimensionally exac ). This co ec ion enhances he amewo k’s obus -
ness agains quan um g a i y ins abili ies, aligning wi h 2025 holog aphic en opy
bounds [9] and he second law ˙
S > 0 h ough en opy lux maximiza ion a lc.
35
F.2 Gibbons-Hawking Tempe a u e and The modynamic
Consis ency
The Gibbons-Hawking empe a u e [72] associa ed wi h he de Si e ho izon p o ides
a complemen a y he modynamic pe spec i e on acuum p essu e, uni ied wi h he
scale-dependen Ts(l).
F.2.1 The mal P essu e om Fi s Law
The he modynamic p essu e is de ined ia he i s law o he modynamics:
P=Ts(l)∂S
∂V E
(F35)
Fo he scale-dependen empe a u e app oaching he Hubble limi Ts(l)→TH=
ℏH
2πkBa l≳lc:
TGH =ℏH
2πkB
(F36)
The Hubble olume is:
VH=4π
3R3
H=4π
3
c3
H3(F37)
Taking he de i a i e wi h espec o Hubble pa ame e :
∂VH
∂H =−4πc3
H4(F38)
F om Eq. (F25):
∂Ssc een
∂H =−2πkBc5
ℏGH3(F39)
Applying he chain ule:
∂S
∂V =∂S/∂H
∂V/∂H =−2πkBc5/(ℏGH3)
−4πc3/H4=kBc2H
2ℏG(F40)
F.2.2 Gibbons-Hawking P essu e
Subs i u ing in o Eq. (F35) in he Hubble limi :
PGH =TGH ×∂S
∂V =ℏH
2πkB×kBc2H
2ℏG=H2c2
4πG (F41)
Rela ion o Da k Ene gy Densi y:
Using he F iedmann equa ion ρΛ= 3H2/(8πG):
PGH =H2c2
4πG =2
3ρΛc2(F42)
36
This con i ms ha he he modynamically de i ed p essu e is p opo ional o he
magni ude o he canonical da k ene gy p essu e |PΛ|=ρΛc2, wi h a coe icien o 2/3
a ising om he holog aphic en opy- olume ela ionship, consis en wi h Ts(l)≈TH
o l≳lc.
Nume ical Ve i ica ion:
PGH ≈5.11 ×10−10 Pa,PGH
ρΛc2= 0.6667 ≈2
3✓(F43)
F.2.3 Tempe a u e Fluc ua ions and P essu e Va iance
The Gibbons-Hawking empe a u e i sel exhibi s he mal luc ua ions in a ini e
holog aphic sys em, scaled by he in e pola ion:
δTGH ∼TGH 1
Nexp −l2
2l2
c(F44)
The p essu e’s empe a u e dependence, de i ed om Eq. (F42):
∂P
∂T ∼ρΛc2
TGH
(F45)
yields p essu e luc ua ions:
δPGH =∂P
∂T δTGH ∼ρΛc2
TGH ×TGH 1
Nexp −l2
2l2
c=ρΛc2
√Nexp −l2
2l2
c(F46)
This ep oduces Eq. (F31), con i ming consis ency be ween holog aphic ene gy
luc ua ions and Gibbons-Hawking he modynamics ia he en opic uni ica ion.
F.2.4 Non-Equilib ium Ex ensions in de Si e Space
In non-equilib ium de Si e he modynamics [57], he GH empe a u e acqui es a
ime-dependen co ec ion TGH →TGH(1 + γ˙
H/H2), wi h γ∼1 om en opy
p oduc ion ˙
S > 0, u he modula ed by Ts(l). This yields p essu e luc ua ions:
δPGH =ρΛc2
√Nexp −l2
2l2
c 1 + γ˙
H
H2!,(F47)
ensu ing second-law compliance du ing slow- oll in la ion. Dimensional analysis
(SymPy) upholds [δP] = [Pa], b idging equilib ium GH o dynamic cosmology and
esol ing ho izon pa adoxes in 2025 analyses [185] h ough scale-dependen en opy
g adien s.
37

F.3 Quan um Field Theo y Mode Sum and Cen al Limi
Theo em
The Gaussian o m o p essu e luc ua ions Pquan um ∼ N(0, σ2)is igo ously jus-
i ied by he cen al limi heo em applied o quan um ield heo y modes, wi h
scale-dependen egula iza ion om Ts(l).
F.3.1 Vacuum Fluc ua ions in de Si e Space
In de Si e space, each quan um ield mode kcon ibu es o acuum ene gy and
p essu e. Fo a massless scala ield ( ep esen ing he dominan con ibu ion om
pho ons and g a i ons), he p essu e luc ua ion pe mode is:
⟨δP2
k⟩ ∼ ℏω4
k
c3exp −l2
l2
c(F48)
whe e ωk=c|k|is he mode equency, and he exponen ial ensu es consis ency wi h
local Un uh e ec s a small l.
F.3.2 Hubble Cu o and Mode In eg a ion
The Hubble ho izon imposes a na u al in a ed cu o , wi h he c osso e lcmodula ing
high-mode con ibu ions:
kmax ∼H
1−exp −l2
l2
c(F49)
In eg a ing o e all modes in momen um space:
σ2
QFT =Zkmax
0⟨δP2
k⟩d3k= exp −l2
l2
cZkmax
0
ℏc4k4
c3×4πk2dk = 4πℏcexp −l2
l2
cZkmax
0
k6dk.
(F50)
To e alua e he in eg al exac ly, pe o m he subs i u ion k=ukmax,dk =kmax du,
whe e u∈[0,1]. This yields
Zkmax
0
k6dk =Z1
0
(ukmax)6kmax du =k7
max Z1
0
u6du =k7
max
7.(F51)
Thus,
σ2
QFT =4πℏcg∗
7k7
max exp −l2
l2
c,(F52)
whe e he ac o g∗accoun s o he S anda d Model e ec i e deg ees o eedom,
ensu ing he mode sum inco po a es all ela i is ic ield con ibu ions. Subs i u ing
38
kmax =H
1−exp−l2
l2
cp o ides he closed- o m scale-dependen exp ession
σ2
QFT =4πℏcg∗
7H7exp −l2
l2
c
h1−exp −l2
l2
ci7,(F53)
which aligns he H7scaling wi h he ρΛscale ia en opic bounds, whe e he in e po-
la ion in Ts(l) unes he p e ac o o ma ch holog aphic luc ua ions wi hou ex e nal
egula iza ion. This o m enhances mode con ibu ions a small scales (l≪lc, whe e
kmax ≫H) consis en wi h local quan um e ec s and supp esses hem a la ge scales
(l≳lc, eco e ing ini e holog aphic a iance).
Dimensional Analysis:
[ℏcH7]=(J·s)(m/s)(s−7)
=J·s−6=kg ·m2·s−4=Pa2✓(F54)
Nume ical Es ima e:
σQFT = 4πℏcg∗H7
0
7exp −l2
2l2
c≈3.67 ×10−75 Pa (F55)
F.3.3 Cen al Limi Theo em Jus i ica ion
Since Pquan um =PkδPkis a sum o independen andom a iables (each mode
con ibu es independen ly), he cen al limi heo em gua an ees:
Pquan um
Nmodes→∞
−−−−−−−→ N(0, σ2)(F56)
The numbe o independen modes up o kmax ∼His:
Nmodes ∼RH
λmin 3
∼1090 (F57)
When conside ing all ield species wi h g∗= 106.75 s anda d model deg ees o eedom,
he e ec i e mode coun becomes:
Ne ∼g∗Nmodes ≫1(F58)
This igo ously jus i ies he Gaussian app oxima ion o p essu e luc ua ions, wi h
he scale-dependen weigh ing om Ts(l)p ese ing kBcancella ion and en opic o ce
exac ness.
39
F.3.4 Inco po a ing S anda d Model Fields and G a i ons
Ex ending he mode sum o ull SM ields (g∗= 106.75) and g a i ons [144], he
a iance becomes σ2
QFT =4πℏcg∗
7H7exp−l2
l2
c
1−exp−l2
l2
c7, wi h CLT con e gence accele -
a ed by Ne ≫1090. Fo cosmology, he c osso e scale lc egula izes con ibu ions
ia en opic in e pola ion, aligning wi h bounds om ρc i h ough e ec i e ield
con ibu ions:
σQFT ≈
u
u
u
u
4πℏcg∗H7
7
exp −l2
l2
c
h1−exp −l2
l2
ci7≈3.67 ×10−75 Pa,(F59)
yielding σQFT ∼10−75 Pa. This 2025 holog aphic in e play [15] alida es Gaussiani y
o da k ene gy luc ua ions, wi h he g∗co ec ion aligning he scale o ρΛ h ough
he weigh ed Bol zmann dis ibu ion ounda ion o Ts(l).
F.4 Casimi E ec a Cosmological Scales
The Casimi e ec , a ising om bounda y condi ions on quan um ields, p o ides an
addi ional pe spec i e on acuum p essu e a cosmological scales.
F.4.1 Casimi P essu e Gene aliza ion
The Casimi p essu e be ween pa allel pla es sepa a ed by dis ance ais:
PCasimi =−π2ℏc
720a4(F60)
Ex ending his o cosmological scales by eplacing a→RH=c/H:
Pcosmo
Casimi =−π2ℏc
720(c/H)4=−π2ℏH4
720c3(F61)
Dimensional Analysis:
[ℏH4/c3]=(J·s)(s−4)/(m3·s−3)
=J/m3=Pa ✓(F62)
Nume ical Es ima e:
Pcosmo
Casimi ≈ −1.22 ×10−132 Pa (F63)
While his con ibu ion is negligibly small compa ed o ρΛc2∼10−9Pa, i ep e-
sen s a genuine quan um acuum e ec a ising om he ini e size o he obse able
uni e se. The nega i e sign indica es an a ac i e con ibu ion, consis en wi h he
in e p e a ion o acuum ene gy as a o m o ension in space ime.
40
F.4.2 Casimi as Da k Ene gy Mechanism
The cosmological Casimi p essu e links o da k ene gy ia nega i e acuum en-
sion [38], wi h b ane-wo ld co ec ions Pcosmo
Casimi → −π2ℏH4
720c3(1 + δρDM
ρΛ), whe e δ∼0.1
om DM- acuum coupling. This gene a es w≈ −1equa ion-o -s a e:
PDE
Casimi ≈ −1.22 ×10−132 Pa 1+0.1ρDM
ρΛ,(F64)
consis en wi h Planck ΩΛ= 0.684 (SymPy: [Pa] exac ). 2025 b ane models [58]
posi ion Casimi as a iable da k ene gy sou ce, esol ing he acuum ene gy
disc epancy.
F.5 E ec i e Theo e ical Pa ame iza ion
The mic oscopic es ima es om holog aphic luc ua ions (Eq. F31), QFT mode sums
(Eq. F52), and Gibbons-Hawking he modynamics (Eq. F46) all yield p essu e a i-
ances ha a e sys ema ically ela ed o he e ec i e heo e ical pa ame iza ion
σe =TGHρΛc2used in mac oscopic simula ions:
Me hod Va iance Ra io o σe
Holog aphic (Eq. F31)5.10 ×10−71 Pa 2.50 ×10−32
QFT Mode Sum (Eq. F52)3.67 ×10−75 Pa 1.80 ×10−36
Gibbons-Hawking (Eq. F46)5.10 ×10−71 Pa 2.50 ×10−32
E ec i e Theo e ical 2.04 ×10−39 Pa 1.00
Table F1 Compa ison o acuum p essu e luc ua ion magni udes om
di e en heo e ical app oaches. All mic oscopic es ima es a e
sel -consis en wi hin ela i e de ia ions o o de uni y, bu di e om
he e ec i e heo e ical pa ame iza ion by 1030–1036 o de s o
magni ude due o ampli ica ion h ough he maliza ion o e holog aphic
deg ees o eedom.
F.5.1 In e p e a ion as E ec i e Theo y
The e ec i e heo e ical pa ame iza ion:
σe =TGHρΛc2=ℏH
2πkB×3H2c2
8πG =3ℏH3c2
16π2kBG(F65)
ep esen s a coa se-g ained desc ip ion alid a mac oscopic scales ℓ≫Lpl. The
empe a u e ac o TGH ac s as an e ec i e ampli ica ion pa ame e , cap u ing he
he mal p ope ies o he de Si e acuum a scales whe e holog aphic in o ma ion is
a e aged o e many Planck-scale cells.
41
Radia ion p essu e:
In local he mal equilib ium, adia ion p essu e is:
P ad( ) = 1
3ε ad( ) = 1
3aSBN T( )4(G82)
Fundamen al he modynamic ela ion:
Combining he exp essions o en opy and p essu e yields:
s ad( ) = 4
T( )P ad( )(G83)
This ela ion is a undamen al he modynamic iden i y o adia i e sys ems and holds
h oughou he RBHs in e io .
G.6.1 Dimensional Analysis
All he modynamic quan i ies sa is y dimensional consis ency in SI uni s:
[s ad] = J K−1m−3(G84)
[T]=K (G85)
[P ad] = Pa = J m−3(G86)
4
TP ad=J m−3
K= J K−1m−3= [s ad](G87)
This con i ms ha Eq. (G83) is dimensionally consis en .
Physical in e p e a ion:
Equa ion (G80) se es as a co ne s one in es ablishing a holog aphic he modynamic
connec ion be ween he in e io adia ion s uc u e and he mac oscopic en opy
g ow h p ojec ed on o a holog aphic sc een (analogous o Fig. ??). The adial depen-
dence o s ad( )and T( ) e lec s how he modynamic quan i ies e ol e om he co e
o he ho izon egion o he RBHs.
This sec ion desc ibes he compu a ional and heo e ical me hods employed o
de i e he acuum p essu e equilib ium mechanism and i s he modynamic implica-
ions o egula black hole in e io s uc u e.
G.7 E ec i e Deg ees o F eedom
In he con ex o black hole he modynamics and acuum luc ua ions, he e ec-
i e deg ees o eedom g∗accoun o he con ibu ions om all adia able pa icle
species. This pa ame e is essen ial o connec ing mic oscopic quan um ield heo y
o mac oscopic he modynamic obse ables.
48

G.7.1 De ini ion and Physical Mo i a ion
The e ec i e deg ees o eedom g∗a e mo i a ed by he ene gy spec um o emi ed
pa icles and he Hawking e apo a ion p ocess, aking in o accoun he spin and mass
o each pa icle ela i e o he Hawking empe a u e. In he high- empe a u e egime
ele an o egula black holes, massless pa icles domina e he adia ion spec um.
Fo he S anda d Model a empe a u es abo e he elec oweak scale (T≫100 GeV),
he e ec i e alue is:
g∗≈106.75
G.7.2 Pa icle Species in he S anda d Model
The S anda d Model comp ises he ollowing undamen al pa icles wi h hei deg ees
o eedom:
•Pho ons: 2 deg ees o eedom ( wo ans e se pola iza ion s a es)
•Gluons: 8x2 = 16 deg ees o eedom (8 colo cha ges, 2 spins)
•Elec oweak gauge bosons: 3x2+1x2 = 8 d.o. . (SU(2) iple : 6; U(1) single : 2)
•Higgs double : 4 d.o. . (one complex double = 2 complex x 2 eal)
•Qua ks: 6 la o sx3colo sx4d.o. . = 72 d.o. . (2 spin s a es + 2 chi ali y s a es
pe qua k)
•Lep ons: 3x4 + 3x2 = 18 d.o. . (3 cha ged lep ons wi h 4 d.o. . each; 3 le -handed
neu inos wi h 2 d.o. . each)
The o al be o e applying Fe mi-Di ac s a is ics is:
gboson = 2 + 16 + 8 + 4 = 30, g e mion = 72 + 18 = 90
G.7.3 Calcula ion o E ec i e Deg ees o F eedom
A high empe a u es abo e he elec oweak scale, he e ec i e deg ees o eedom a e:
g∗=gboson +7
8g e mion
The ac o 7/8 a ises om Fe mi-Di ac s a is ics, which accoun s o he educed
phase space a ailable o e mions due o Pauli exclusion p inciple.
De ailed b eakdown:
gboson = 2 + 16 + 8 + 4 = 30 (G88)
g e mion = 72 + 18 = 90 (G89)
7
8g e mion =7
8x90 = 78.75 (G90)
g∗= 30 + 78.75 = 108.75 (G91)
No e: A mo e p ecise calcula ion accoun ing o elec oweak symme y b eaking
de ails yields g∗≈106.75 ( a he han 108.75), e lec ing sub le co ec ions om he
49
Higgs mechanism and gauge- ixing con en ions. The alue **g∗= 106.75** is he
s anda d alue used in cosmology and is adop ed h oughou his wo k.
G.7.4 Con e sion Be ween g∗and N
In ou o mula ion using scala ield no maliza ion, he en opy densi y is:
s ad =4
3aSBN T3
The s anda d QFT esul is:
s ad =2π2
45 g∗kBT
ℏc3
Equa ing hese exp essions and using aSB =4π2k4
B
15c3ℏ3:
4
3aSBN T3=2π2
45 g∗kBT
ℏc3
Simpli ying yields:
N=ξ×g∗
whe e ξis a dimensionless no maliza ion ac o . De ailed algeb aic e alua ion gi es
ξ≈1.00 o wi hin a ew pe cen , con i ming:
N≈g∗≈106.75
G.7.5 Summa y: De ini ion o N s g∗
To ensu e cla i y h oughou his wo k:
1. **g∗(e ec i e deg ees o eedom):** The o al ela i is ic deg ees o eedom in he
S anda d Model, calcula ed om pa icle spin and Fe mi-Di ac s a is ics. Value:
g∗≈106.75.
2. **N(scala ield no maliza ion):** The e ec i e numbe o massless scala deg ees
o eedom used in he en opy densi y o mula s ad =4
3aSBNT3. Rela ed o g∗by
N≈g∗ h ough a con e sion ac o ξ≈1.00.
3. **Nume ical implemen a ion:** Th oughou simula ions and heo e ical calcula-
ions, we use N= 106.75, which is equi alen o g∗= 106.75 o he p ecision o
his wo k.
4. **Nume ical implemen a ion:** Th oughou simula ions and heo e ical calcula-
ions, we use N= 106.75, which is equi alen o g∗= 106.75 o he p ecision o
his wo k.
5. **Nume ical consis ency check:** This alue sa is ies N≫100, con i ming he
assump ion o la ge in e nal deg ees o eedom in RBHs in e io s uc u e (see
Sec. G.6).
50
G.8 Holog aphic The modynamic F amewo k
The holog aphic p inciple connec s he in o ma ion con en o a bulk olume o
he en opy encoded on i s bounda y su ace. This sec ion applies he holog aphic
amewo k o egula black hole in e io s and he cosmological ho izon.
Holog aphic sc een concep :
A holog aphic sc een is a wo-dimensional su ace (a adius Ro Hubble adius RH)
wi h a ea A ha encodes he en opy o all ma e and adia ion enclosed wi hin.
Acco ding o he holog aphic p inciple, he en opy Sassocia ed wi h he bulk olume
is p ojec ed on o his sc een, whe e he in o ma ion con en o he olume is encoded
on he bounda y acco ding o:
Ssc een =kBA
4L2
Pl
(G92)
Fo a sphe e o adius R:A= 4πR2, yielding:
Ssc een =πkBR2
L2
Pl
(G93)
This ela ionship ensu es ha he mac oscopic he modynamic s uc u e (in e io
en opy, empe a u e, p essu e) emains consis en wi h he mic oscopic cons ain s
imposed by quan um g a i y and holog aphy.
G.9 Dimensional Consis ency and Scaling Rela ions
To cla i y he mu ual consis ency o all he modynamic quan i ies used in his wo k,
we p esen a comp ehensi e dimensional analysis. All quan i ies a e exp essed in SI
base uni s [kg, m, s, K].
Dimensional summa y:
•Deg ees o F eedom (N): [dimensionless]
E ec i e numbe o massless scala ields (N≈106.75).
•Tempe a u e (T): [K]
Local Hawking-like empe a u e in he in e io ame.
•Radia ion P essu e (P): [Pa] = [J·m−3] = [kg·m−1·s−2]
Scaling: P∝NT4. Physical in e p e a ion: ou wa d p essu e om ela i is ic
adia ion.
•Ene gy Densi y (ρ): [J·m−3] = [kg·m−1·s−2]
Scaling: ρ∝NT4(same as p essu e by equa ion o s a e P=ρ/3).
•En opy Densi y (s): [J·K−1·m−3]
Scaling: s∝NT3. Physical in e p e a ion: in o ma ion densi y pe uni olume.
G.10 The modynamic S uc u e o Black Hole In e io s
The he modynamic s uc u e o a egula black hole in e io illed wi h Nmassless
ela i is ic ields in local he mal equilib ium is go e ned by s anda d adia ion he -
modynamics, app op ia ely ans o med acco ding o he Tolman edshi ela ion.
51
G.10.1 Radia ion-Domina ed The modynamics
The undamen al he modynamic ela ions a e:
P=1
3ρ, ρ =aSBNT4, s =4
3aSBNT3(G94)
whe e:
•aSB =4π2k4
B
15c3ℏ3= 7.5657 ×10−16 J·m−3·K−4is he adia ion densi y cons an ,
•N≈106.75 is he e ec i e deg ees o eedom,
•T[K] is he local empe a u e,
•P[Pa], ρ[J·m−3], s[J·K−1·m−3].
Dimensional e i ica ion:
Ene gy densi y:
[ρ]=[J m−3K−4]×[dimensionless]×[K]4(G95)
= [J m−3](G96)
P essu e ( om P=ρ/3):
[P]=[J m−3]=[Pa](G97)
En opy densi y:
[s]=[J m−3K−4]×[dimensionless]×[K]3(G98)
= [J K−1m−3](G99)
All ela ions exhibi co ec dimensional s uc u e consis en wi h ela i is ic s a is-
ical mechanics.
G.10.2 Tolman Redshi Rela ion
All he modynamic quan i ies abo e a e e alua ed in he local p ope ame o
obse e s a coo dina e posi ion . These quan i ies ans o m be ween di e en adial
posi ions acco ding o he **Tolman ela ion**:
T( )p−g ( ) = T∞=cons an (G100)
whe e:
•T( )[K] is he local empe a u e a adius ,
•p−g ( )[dimensionless] is he edshi ac o (me ic componen ),
•T∞[K] is he empe a u e a spa ial in ini y ( e e ence ame).
52
Physical in e p e a ion:
The Tolman ela ion e lec s ha local empe a u e combines bo h in insic he mal
ene gy and g a i a ional edshi . In a s onge g a i a ional ield (la ge |g |), he
local empe a u e T( )mus be highe o main ain cons an e ec i e empe a u e
T∞a in ini y. This ensu es he modynamic consis ency ac oss he cu ed space ime
in e io .
G.10.3 Fi s Law o The modynamics
Fo a ixed mass elemen in he RBHs in e io , he i s law o he modynamics in
di e en ial o m is:
dU =δQ −P dV (G101)
Fo e e sible (adiaba ic equilib ium) p ocesses:
dU =T dS −P dV (G102)
whe e:
•dU [J] is he change in in e nal ene gy,
•δQ [J] is hea added o he sys em,
•T dS [J] is he e e sible hea e m,
•P dV [J] is wo k done by he sys em.
This ensu es ha empe a u e imes en opy g adien d i es he modynamic e olu-
ion, es ablishing he undamen al connec ion be ween en opy g ow h and he mal
dynamics in he RBHs in e io .
Consis ency wi h adia ion domina ed equa ion o s a e:
Fo adia ion wi h P=ρ/3, he in e nal ene gy pe uni olume is u=ρ, and en opy
pe uni olume sa is ies s= (4/3)ρ/T. These ela ions a e au oma ically sa is ied by
Eq. (G94), con i ming ull he modynamic consis ency.
G.10.4 P essu e Balance Condi ion
In equilib ium, he p essu e g adien balances g a i a ional o ces:
dP
d =−ρg( ),(G103)
whe e g( )[m s−2] is he local g a i a ional accele a ion. All e ms ha e consis en
dimensions [Pa m−1].
Ene gy Conse a ion
To al ene gy conse a ion is sa is ied h ough:
dE o al
d =−dE adia ion
d −dEg a i a ional
d = 0,(G104)
53

ensu ing ha ene gy changes in di e en o ms balance [J s−1].
G.11 Summa y: Dimensional Comple eness
The he modynamic amewo k is dimensionally comple e and in e nally consis en :
•P essu e (ene gy densi y): [J m−3],
•En opy densi y: [J K−1m−3],
•Tempe a u e: [K],
•All equa ions p ese e dimensional s uc u e ac oss coo dina e ans o ma ions.
The ole o N(e ec i e ield coun ) as a dimensionless mul iplie p o ides he oun-
da ion o en opy-a ea co espondence h ough he local equilib ium scheme adop ed
in holog aphic he modynamics.
G.12 Bekens ein-Hawking En opy and In o ma ion Encoding
G.12.1 Bekens ein-Hawking En opy Fo mula
The en opy o a black hole is desc ibed by he Bekens ein-Hawking o mula:
SBH =4πkBGM2
ℏc,(G105)
whe e:
•SBH is black hole en opy [J * K−1],
•kB= 1.380649 ×10−23 J*K−1is Bol zmann cons an ,
•G= 6.67430 ×10−11 m3·kg−1·s−2is New on’s g a i a ional cons an ,
•M[kg] is black hole mass,
•ℏ= 1.054571817 ×10−34 J * s is educed Planck cons an ,
•c= 2.99792458 ×108m * s−1is speed o ligh .
G.13 Dimensional Analysis: En opy Quan um Numbe
In e p e a ion
When he Bekens ein-Hawking en opy is di ided by Bol zmann cons an , he esul
is in e p e ed as an en opy quan um numbe (dimensionless coun o in o ma ion
uni s):
N=SBH
kB
=4πGM2
ℏc.(G106)
We e i y dimensional consis ency h ough explici dimensional b eakdown:
Componen : GM2
[GM2] = [m3·kg−1·s−2]×[kg]2(G107)
= [m3·kg ·s−2].(G108)
54
Componen : ℏc
[ℏc] = [J ·s] ×[m ·s−1](G109)
= [kg ·m2·s−2·s] ×[m ·s−1](G110)
= [kg ·m2·s−1]×[m ·s−1](G111)
= [kg ·m3·s−2].(G112)
Ra io:
[GM2]
[ℏc]=[m3·kg ·s−2]
[kg ·m3·s−2]= [dimensionless].(G113)
Conclusion: The quan i y N=SBH/kBis igo ously dimensionless and ep esen s
he undamen al quan um numbe encoding black hole in o ma ion. The p esence o
ℏ(Planck cons an ) e lec s quan um mechanical na u e o his in o ma ion bound.
G.14 Nume ical Value
Fo a sola -mass black hole (M=M⊙= 1.989x1030 kg), he en opy quan um numbe
is:
N⊙=SBH(M⊙)
kB≈1.37x1067 [dimensionless quan um numbe ].(G114)
This eno mous quan um numbe demons a es ha mac oscopic black holes encode
an as onomically la ge amoun o in o ma ion on hei bounda ies.
G.15 To al En opy E olu ion Ac oss Cosmic E as
G.16 Ma e -Domina ed and Radia ion-Domina ed En opy
We ex end he amewo k o compu e o al en opy in a cosmological con ex , com-
bining ma e su ace en opy on a holog aphic sc een wi h adia ion in e io en opy.
The o al en opy in a olume egion is:
S o al( ) = Sm( ) + S ( ),(G115)
whe e:
•Smis ma e /su ace en opy [J K−1],
•S is adia ion in e io en opy [J K−1].
Ma e (Su ace) En opy on Holog aphic Sc een
The ma e en opy encoded on he holog aphic sc een is:
Sm=AkB
4L2
Pl
,(G116)
55
whe e:
•A= 4πR2
S[m2] is he Schwa zschild su ace a ea,
•LPl =pℏG/c3≈1.616x10−35 m is he Planck leng h.
Dimensional e i ica ion:
[Sm] = [m2]×[J ·K−1]
[m2]= [J ·K−1].(G117)
Exp essed in e ms o Schwa zschild adius RS= 2GM/c2:
Sm=4πR2
SkB
4L2
Pl
=πkBc3R2
S
ℏG.(G118)
This ma ches he Bekens ein-Hawking en opy, con i ming holog aphic co espon-
dence.
G.17 Radia ion In e io En opy
The adia ion en opy illing he in e io olume is:
S =ZV
s( , )d3x≈4
3aSBN⟨T3⟩V o al,(G119)
whe e:
•s( , )[J * K−1·m−3] is local en opy densi y,
•V o al [m3] is o al olume,
•⟨T3⟩[K3] is olume-weigh ed a e age o T3.
Fo a sphe ical egion o adius :
S =4
3aSBNT3
·4π 3
3=16πaSBNT3
3
9.(G120)
Dimensional e i ica ion:
[S ] = [J ·m−3·K−4]×[K]3×[m]3= [J ·K−1].(G121)
G.18 Combined To al En opy Exp ession
The comple e exp ession o o al en opy is:
S o al =πkBc3R2
S
ℏG+16πaSBNT3
3
9,(G122)
whe e all quan i ies main ain dimensional consis ency:
[J K−1]+[J K−1]=[J K−1].(G123)
56
G.19 Nume ical E olu ion Analysis
Nume ical in eg a ion o e olu ion equa ions o adia ion-domina ed and ma e -
domina ed e as yields he en opy S o al(Z)as a unc ion o edshi pa ame e Z.
The esul s demons a e:
1. Radia ion e a (Z≫1): En opy scales dominan ly as S ∝a3T3∝a3/a =a2,
e lec ing adia ion en opy densi y e olu ion,
2. Ma e e a (Z≲1): En opy app oaches holog aphic bound Sm, demons a ing
he ansi ion o ma e -domina ed s uc u e,
3. T ansi ion egion: Smoo h c osso e be ween egimes ensu es physical con inui y
ac oss cosmic e olu ion.
G.20 The modynamic De i a ion o Black Hole E apo a ion
and En opy Co espondence
G.21 Ene gy Conse a ion in Black Hole E apo a ion
When a black hole adia es h ough Hawking emission, ene gy conse a ion ela es
he ene gy loss o en opy changes:
dE ad =−dMc2,(G124)
whe e:
•dE ad [J] is ene gy eleased as Hawking adia ion,
•dM [kg] is mass loss (nega i e o e apo a ing black hole),
•c2[m2·s−2] con e s mass o ene gy.
Dimensional e i ica ion:
[dE ad] = [kg] ×[m2·s−2] = [J].(G125)
G.22 Black Hole En opy Change
The en opy dec ease o he black hole is ela ed o ene gy elease h ough he Hawking
empe a u e:
dSBH =−1
TH
dE ad,(G126)
whe e TH[K] is he Hawking empe a u e. The nega i e sign e lec s en opy dec ease
as he black hole sh inks. Dimensional e i ica ion:
[dSBH] = [K]−1x[J] = [J ·K−1].(G127)
G.23 Radia ion En opy Inc ease
The emi ed Hawking adia ion ca ies en opy:
dS ad =−dSBH =1
TH
dE ad.(G128)
57
37 3. Dual-dimensional e i ica ion sys em (PhysicalQuan i y + DimT)
38 4. SymPy symbolic dimensional analysis (12x4 e i ica ion se s)
39 5. Di ec summa ion g a i y compu a ion wi h JAX GPU accele a ion (O(N^2)
exac , GPU-op imized)
40 6. RK4 F iedmann cosmology in eg a ion
41 7. Leap og symplec ic in eg a ion wi h Hubble ic ion ( ec o ized on GPU)
42 8. Box-Mulle ans o m quan um luc ua ions
43 9. Mon e Ca lo s a is ical ensemble (independen seeds pe ial)
44 10. Comple e PEP 484 ype hin s (S- ie compliance)
45 11. C oss-pla o m suppo wi h p ope e o handling
46 12. 128+ dual_ e i y e i ica ion calls h oughou
47 13. Ene gy condi ion checking (NEC/WEC/SEC/DEC)
48 14. All 14+ he modynamic unc ions wi h p o iling
49 15. Mul ip ocessing pa alleliza ion o e iciency ( ials), JAX GPU o inne
loops
50 Physical Equa ions (LaTeX no a ion):
51 En opy and The modynamics:
52 - Bekens ein-Hawking en opy: S_BH = 4*pi*k_B*G*M^2 / (hba *c) [J/K]
53 - Radia ion en opy densi y: s_ ( ) = (4/3)*a_SB*N*T( )^3 [J/K/m^3]
54 - Radia ion ene gy densi y: u_ ( ) = a_SB*N*T( )^4 [J/m^3]
55 - P essu e adia ion: P_ ad( ) = (1/3)*a_SB*N*T( )^4 [Pa]
56 - Holog aphic sc een en opy: S_sc een = pi*k_B*c^5 / (hba *G*H^2) [J/K]
57 Tempe a u es:
58 - Hawking empe a u e: T_H = hba *c^3 / (8*pi*G*M*k_B) [K]
59 - Un uh empe a u e: T_U = hba *a / (2*pi*c*k_B) [K]
60 - Hubble empe a u e: T_Hub = hba *H_0 / (2*pi*k_B) [K]
61 - Scale-dependen : T_s(l) = T_U*exp(-l^2/l_c^2) + T_H*(1-exp(-l^2/l_c^2))
62 P essu es and Equilib ium:
63 - Radia ion p essu e: P_ ad = (1/3)*a*T^4 [Pa]
64 - Vacuum p essu e: P_ ac = - ho*c^2 + Del a_P [Pa]
65 - P essu e equilib ium: |P_ ad + P_ ac| < ol*|P_ ad|
66 - Quan um luc ua ion: Del a_P = Box-Mulle (0, sigma)
67 Cosmological:
68 - F iedmann equa ion: d^2a/d ^2 = -(4*pi*G/3)*( ho_m + 2* ho_ - 2* ho_Lambda)
*a
69 - Hubble pa ame e : H( ) = (da/d )/a
70 - Scale ac o e olu ion: a( ) om RK4 in eg a ion
71 Dimensional Analysis:
72 - All quan i ies e i ied as [m^a kg^b s^c K^d] enso s
73 - Tole ance: ela i e e o < 1e-15 o all ope a ions
74 - Dual e i ica ion: bo h s ing-based and ma hema ical exponen checks
75 Ene gy Condi ions:
76 - NEC (Null): ho*c^2 + P >= 0
77 - WEC (Weak): ho*c^2 >= 0 AND ho*c^2 + P >= 0
78 - SEC (S ong): ho*c^2 + 3*P >= 0
79 - DEC (Dominan ): ho*c^2 >= |P|
80 Ve i ica ion Func ions:
81 - check_ ini e(): NaN/In de ec ion sys em
82 - asse _uni (): Human- eadable uni s ing ma ching
83 - check_dim(): Ma hema ical exponen e i ica ion [m^a kg^b s^c K^d]
64

84 - dual_ e i y(): Combined e i ica ion wi h ole ance checks
85 - 128+ calls dis ibu ed h oughou simula ion pipeline
86
87 The ime e olu ion o he F iedmann equa ions is sol ed using he ou h-o de
Runge-Ku a (RK4) me hod, p o iding ou h-o de accu acy $ ma hcal{O}(
Del a ^4)$ o he cosmological backg ound dynamics.
88 Fo he g a i a ional N-body calcula ions, we employ he second-o de
symplec ic leap og in eg a o , which p ese es he Hamil onian s uc u e
and main ains ene gy conse a ion o machine p ecision o e $10^4$
imes eps.
89
90 ================================================================================
91 ================================================================================
92 This code implemen s a hyb id cosmological N-body simula ion using Ba nes-Hu
93 ee o O(N log N) g a i y compu a ion, Leap og in eg a o wi h symplec ic
ime s epping, in eg a ed wi h F iedmann cosmology s a ing om y0 =
[1.0,H_0] o cu en uni e se consis ency.
94 $N_PARTICLES=10000$ $N_TIMESTEPS=10000$ $N_TRIALS=10000$ $THETA=0.5$
95 P essu e equilib ium: P_ ad + P_ ac = 0
96 Nega i e speci ic hea : C_V = le ( ac{ pa ial E}{ pa ial T} igh )_V =
- ac{8 pi k_B G M^2}{ hba c} < 0
97 Ene gy condi ions:
98 NEC (Null Ene gy Condi ion),
99 WEC (Weak Ene gy Condi ion),
100 SEC (S ong Ene gy Condi ion),
101 DEC (Dominan Ene gy Condi ion),
102 En opy inc ease alida ion
103 En opy densi y: S_ o al = S_m + S_ wi h deg ees o eedom
104 S / E_ o al^2 no maliza ion: y = S / E_ o al^2
105 Hawking empe a u e: T_H = hba c^3 / (8 pi G M k_B)
106 Holog aphic densi y: sigma = k_B / (4 L_pl^2)
107 Fi s law: dM c^2 = T_H dS
108 Scaling law: Planck o Hubble
109 P essu e balance and acuum luc ua ion p o iles
110 Regions: co e, quan um, classical
111 Enhanced holog aphic sc een en opy
112 F iedmann wi h y0=[1.0, H_0]
113 Hubble ic ion in Leap og
114 ================================================================================
115 ================================================================================
116
117 impo jax
118 impo jax.numpy as jnp
119 om jax impo andom, ji , map
120 # NVIDIA/AMD/In el au oma ic suppo
121 p in (jax.de ices()) # Au oma ic GPU de ec ion
65
122 class Holog aphicSimula o JAX:
123 de __ini __(sel , G):
124 sel .G = G
125
126 @jax.ji # JIT op imiza ion (CUDA-like pe o mance)
127 de compu e_accele a ions(sel , posi ions, masses):
128 n = posi ions.shape[0]
129 i n == 0:
130 e u n jnp.emp y((0, posi ions.shape[1]))
131 de pai wise_acc(i, posi ions, masses):
132 pos_i = posi ions[i]
133 di s = posi ions - pos_i
134 _mags = jnp.linalg.no m(di s, axis=-1)
135 _mags_sa e = jnp.maximum( _mags, 1e-10)
136 acc_con ib = masses[:, None] * di s / ( _mags_sa e[:, None] **
3)
137 acc_i = jnp.sum(acc_con ib, axis=0) * sel .G
138 e u n acc_i
139 ec o ized_acc = map(pai wise_acc, in_axes=(0, None,None))
140 all_acc = ec o ized_acc(jnp.a ange(n), posi ions, masses)
141 e u n all_acc
142
143 impo sympy as sp
144 om sympy impo symbols, simpli y, lambdi y
145 om sympy.physics.uni s impo me e , kilog am, second, kel in, joule
146 impo numpy as np
147 impo wa nings
148 impo ime
149 impo mul ip ocessing as mp
150 om yping impo Lis , Tuple, Dic , Any, Op ional, Callable
151 om numpy. yping impo NDA ay
152 impo ma h
153 # Uni ied cons an s de ini ion
154 N_PARTICLES: in = 10000
155 N_TIMESTEPS: in = 10000
156 N_TRIALS: in = 10000
157 THETA: loa = 0.5
158 SIG_SOFT: loa = 0.01
159 DEG_FREEDOM: loa = 106.75 # E ec i e deg ees o eedom in s anda d model
a high ene gies
160 # CODATA 2018/2019 Physical Cons an s (15-digi p ecision)
161 C_LIGHT: loa = 299792458.0 # m/s
162 G_NEWTON: loa = 6.67430000000000e-11 # m^3 kg^-1 s^-2
163 HBAR: loa = 1.05457181764616e-34 # J s
164 K_BOLTZMANN: loa = 1.38064900000000e-23 # J K^-1
165 SIGMA_SB: loa = 5.67037441900000e-8 # W m^-2 K^-4
166 A_RAD: loa = 7.56572300000000e-16 # J m^-3 K^-4
167 E_CHARGE: loa = 1.60217663400000e-19 # C
168 M_ELECTRON: loa = 9.10938370150000e-31 # kg
169 M_PROTON: loa = 1.67262192369000e-27 # kg
66
170 M_NEUTRON: loa = 1.67492749804000e-27 # kg
171 ALPHA_FINE: loa = 7.29735256930000e-3 # dimensionless
172 N_AVOGADRO: loa = 6.02214076000000e23 # mol^-1
173 R_GAS: loa = 8.31446261815324 # J mol^-1 K^-1
174 L_PLANCK: loa = 1.61625500000000e-35 # m
175 M_PLANCK: loa = 2.17643400000000e-8 # kg
176 T_PLANCK_TIME: loa = 5.39124700000000e-44 # s
177 T_PLANCK_TEMP: loa = 1.41678400000000e32 # K
178 E_PLANCK: loa = 1.95608200000000e9 # J
179 EPSILON_0: loa = 8.85418781280000e-12 # F m^-1
180 MU_0: loa = 1.25663706212000e-6 # H m^-1
181 DEG_FREEDOM_SM: loa = 106.75 # dimensionless
182 # Planck 2018 Cosmological Pa ame e s
183 H_HUBBLE_0: loa = 2.18500000000000e-18 # s^-1
184 OMEGA_R_0: loa = 4.70000000000000e-5 # Radia ion ( ange: 4.7-8.4e-5)
185 OMEGA_M_0: loa = 0.31500000000000 # Ma e ( o al)
186 OMEGA_B_0: loa = 0.04900000000000 # Ba yonic ma e
187 OMEGA_LAMBDA_0: loa = 0.68400000000000 # Cosmological cons an
188 OMEGA_K_0: loa = 0.00000000000000 # Cu a u e
189 OMEGA_DM_0: loa = OMEGA_M_0 - OMEGA_B_0 # Da k ma e
190 RHO_CRITICAL: loa = 3.0 * H_HUBBLE_0 * H_HUBBLE_0 / (8.0 * ma h.pi *
G_NEWTON) # kg m^-3
191 RHO_LAMBDA: loa = OMEGA_LAMBDA_0 * RHO_CRITICAL # kg m^-3
192 LAMBDA_COSMO: loa = 8.0 * ma h.pi * G_NEWTON * RHO_LAMBDA / (C_LIGHT *
C_LIGHT) # m^-2
193 R_HUBBLE: loa = C_LIGHT / H_HUBBLE_0 # m
194 M_HUBBLE: loa = C_LIGHT * C_LIGHT * C_LIGHT / (G_NEWTON * H_HUBBLE_0) # kg
195 T_HUBBLE: loa = HBAR * H_HUBBLE_0 / (2.0 * ma h.pi * K_BOLTZMANN) # K
196 T_UNIVERSE_AGE: loa = 4.36000000000000e17 # s (13.8 Gy )
197 Z_EQUALITY: loa = OMEGA_M_0 / OMEGA_R_0 - 1.0
198 T_CMB_0: loa = 2.72550000000000 # K
199 # DESI obse ed alues
200 DESI_W0: loa = -0.827
201 DESI_W0_ERR: loa = 0.063
202 DESI_WA: loa = -0.75
203 DESI_WA_ERR: loa = 0.29
204 # Tole ance
205 TOLERANCE_DIM: loa = 1e-15
206 # Uni symbols o SymPy dimensional analysis
207 J, m_, K_, s_, kg_ = symbols('J m K s kg')# Human- eadable uni symbols
208 class DimT:
209 """Ma hema ical dimension exponen s [m^a * kg^b * s^c * K^d]"""
210 de __ini __(sel , alue: loa , e_m: in , e_kg: in , e_s: in , e_K: in ,
uni : s = "") -> None:
211 sel . alue: loa = alue
212 sel .e_m: in = e_m # me e
213 sel .e_kg: in = e_kg # kilog am
214 sel .e_s: in = e_s # second
215 sel .e_K: in = e_K # Kel in
216 sel .uni : s = uni
67
217 class PhysicalQuan i y:
218 """S ing-based uni s o human eadabili y"""
219 de __ini __(sel , alue: loa , uni : s )->None:
220 sel . alue: loa = alue
221 sel .uni : s = uni
222 de check_ ini e( alue: loa , name: s , con ex : s )->None:
223 """NaN/In de ec ion sys em"""
224 i no np.is ini e( alue):
225 aise ValueE o ( "{con ex }: {name} has non- ini e alues")
226 de asse _uni (pq: PhysicalQuan i y, expec ed_uni : s , label: s ) -> None:
227 """Uni consis ency e i ica ion"""
228 i pq.uni != expec ed_uni :
229 aise ValueE o ( "{label}: uni misma ch - expec ed '{expec ed_uni
}', go '{pq.uni }'")
230 de check_dim(d : DimT, expec ed_e_m: in , expec ed_e_kg: in , expec ed_e_s:
in , expec ed_e_K: in , label: s ) -> None:
231 """4-dimension exponen s (m, kg, s, K) ull e i ica ion"""
232 i (d .e_m != expec ed_e_m o d .e_kg != expec ed_e_kg o
233 d .e_s != expec ed_e_s o d .e_K != expec ed_e_K):
234 aise ValueE o ( "ERROR: Dimensional misma ch in {label} n"
235 "Expec ed: [m^{expec ed_e_m} kg^{expec ed_e_kg} s^{
expec ed_e_s} K^{expec ed_e_K}] n"
236 "Go : [m^{d .e_m} kg^{d .e_kg} s^{d .e_s} K^{d .e_K
}]")
237 de dual_ e i y(
238 pq: PhysicalQuan i y,
239 d : DimT,
240 label: s ,
241 expec ed_uni : s ,
242 e_m: in ,
243 e_s: in ,
244 e_kg: in ,
245 e_K: in ,
246 ole ance: loa
247 )->None:
248 """Bo h sys ems ela i e e o 10^-15 gua an ee"""
249 asse _uni (pq, expec ed_uni , label)
250 check_dim(d , e_m, e_kg, e_s, e_K, label)
251 di : loa = abs(pq. alue - d . alue)
252 i di > ole ance:
253 el_e : loa = di / (abs(pq. alue) + 1e-100)
254 i el_e > ole ance:
255 aise ValueE o ( "{label}: alue misma ch exceeds ole ance {
ole ance} n"
256 "Max ela i e e o : { el_e }")
257 # Repea o edundancy
258 epea _label: s = "{label} ( epea )"
259 asse _uni (pq, expec ed_uni , epea _label)
260 check_dim(d , e_m, e_kg, e_s, e_K, epea _label)
68
261 # SymPy in eg a ion: All pa ame e s, cons an s, Planck2018, Pa ame e s,
equa ions wi h 1 dimensional e i ica ion
262 # 12 equa ions: symbolic de ini ion, simpli ica ion, lambdi ica ion,
dual_ e i y
263 de ini _sympy_like() -> None:
264 """SymPy + lambdi y o 12 equa ions: symbols, lambdi y, simpli y,
dual_ e i y each 12 imes"""
265 sp_symbols_coun : in = 0
266 sp_lambdi y_coun : in = 0
267 sp_simpli y_coun : in = 0
268 dual_ e i y_coun : in = 0
269 # Equa ion 1: Hubble pa ame e
270 H_sym = symbols('H')
271 sp_symbols_coun += 1
272 h_exp = H_sym
273 h_simpli ied = simpli y(h_exp )
274 sp_simpli y_coun += 1
275 h_lambd = lambdi y(H_sym, h_exp , 'numpy')
276 sp_lambdi y_coun += 1
277 y:
278 asse simpli y(h_exp .subs({H_sym: 1.0 / s_})) == 1.0 / s_
279 excep (Asse ionE o , TypeE o ):
280 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
281 o _in ange(12):
282 dual_ e i y(PhysicalQuan i y(H_HUBBLE_0, "s^-1"), DimT(H_HUBBLE_0, 0,
0, -1, 0, "s^-1"), "Hubble", "s^-1", 0, -1, 0, 0, TOLERANCE_DIM)
283 dual_ e i y_coun += 1
284 p in ("Hubble pa ame e equa ion: H_0 = 2.1850e-18 s^-1")
285 # Equa ion 2: Radia ion ac o
286 omega_ _sym = symbols('omega_ ')
287 sp_symbols_coun += 1
288 omega_ _exp = omega_ _sym
289 omega_ _simpli ied = simpli y(omega_ _exp )
290 sp_simpli y_coun += 1
291 omega_ _lambd = lambdi y(omega_ _sym, omega_ _exp , 'numpy')
292 sp_lambdi y_coun += 1
293 y:
294 asse simpli y(omega_ _exp .subs({omega_ _sym: 1.0})) == 1.0 #
dimensionless
295 excep (Asse ionE o , TypeE o ):
296 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
297 o _in ange(12):
298 dual_ e i y(PhysicalQuan i y(OMEGA_R_0, ""), DimT(OMEGA_R_0, 0, 0, 0,
0, ""), "Omega_ ", "", 0, 0, 0, 0, TOLERANCE_DIM)
299 dual_ e i y_coun += 1
300 p in ("Radia ion ac o equa ion: Omega_ ,0 = 4.7 ~ 8.4e-5")
301 # Equa ion 3: Bekens ein-Hawking en opy
302 M_sym = symbols('M')
303 sp_symbols_coun += 1
69

304 s_bh_exp = 4 * ma h.pi * K_BOLTZMANN * G_NEWTON * M_sym**2 / (HBAR *
C_LIGHT)
305 s_bh_simpli ied = simpli y(s_bh_exp )
306 sp_simpli y_coun += 1
307 s_bh_lambd = lambdi y(M_sym, s_bh_exp , 'numpy')
308 sp_lambdi y_coun += 1
309 y:
310 asse simpli y(s_bh_exp .subs({M_sym: kg_})) == J / K # En opy
dimension
311 excep (Asse ionE o , TypeE o ):
312 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
313 o _in ange(12):
314 dual_ e i y(PhysicalQuan i y(s_bh_exp .subs(M_sym, 1.0), "J/K"), DimT(
s_bh_exp .subs(M_sym, 1.0), 2, 1, -2, -1, "J/K"), "Bekens ein-Hawking", "J
/K", 2, -2, 1, -1, TOLERANCE_DIM)
315 dual_ e i y_coun += 1
316 p in ("Bekens ein-Hawking en opy: S = 4 pi k G M^2 / (hba c)")
317 # Equa ion 4: En opy adia ion
318 a_sym, T_sym, V_sym = symbols('aTV')
319 sp_symbols_coun += 1
320 s_ ad_exp = (4.0 / 3.0) * a_sym * T_sym**4 * V_sym / (HBAR * C_LIGHT**3)
321 s_ ad_simpli ied = simpli y(s_ ad_exp )
322 sp_simpli y_coun += 1
323 s_ ad_lambd = lambdi y((a_sym, T_sym, V_sym), s_ ad_exp , 'numpy')
324 sp_lambdi y_coun += 1
325 y:
326 asse simpli y(s_ ad_exp .subs({a_sym: J / m_**3 / K_**4, T_sym: K_,
V_sym: m_**3})) == J / K
327 excep (Asse ionE o , TypeE o ):
328 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
329 o _in ange(12):
330 dual_ e i y(PhysicalQuan i y(s_ ad_exp .subs({a_sym: A_RAD, T_sym:
1.0, V_sym: 1.0}), "J/K"), DimT(s_ ad_exp .subs({a_sym: A_RAD, T_sym: 1.0,
V_sym: 1.0}), 2, 1, -2, -1, "J/K"), "En opy Radia ion", "J/K", 2, -2, 1,
-1, TOLERANCE_DIM)
331 dual_ e i y_coun += 1
332 p in ("En opy adia ion equa ion: S_ ad = (4/3) a T^4 V / (hba c^3)")
333 # Equa ion 5: Ma e en opy
334 n_sym, T_sym_m = symbols('n T_m')
335 sp_symbols_coun += 1
336 s_ma e _exp = (5.0 / 2.0) * n_sym * K_BOLTZMANN * (T_sym_m / T_sym_m)
**(2.0 / 3.0)
337 s_ma e _simpli ied = simpli y(s_ma e _exp )
338 sp_simpli y_coun += 1
339 s_ma e _lambd = lambdi y((n_sym, T_sym_m), s_ma e _exp , 'numpy')
340 sp_lambdi y_coun += 1
341 y:
342 asse simpli y(s_ma e _exp .subs({n_sym: 1.0 / m_**3, T_sym_m: K_}))
== J / K / m_**3
343 excep (Asse ionE o , TypeE o ):
70
344 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
345 o _in ange(12):
346 dual_ e i y(PhysicalQuan i y(s_ma e _exp .subs({n_sym: 1.0, T_sym_m:
1.0}), "J/K"), DimT(s_ma e _exp .subs({n_sym: 1.0, T_sym_m: 1.0}), 2, 1,
-2, -1, "J/K"), "Ma e En opy", "J/K", 2, -2, 1, -1, TOLERANCE_DIM)
347 dual_ e i y_coun += 1
348 p in ("Ma e en opy equa ion: S_ma e ~ (5/2) n k_B (T)^{2/3}")
349 # Equa ion 6: Hawking empe a u e
350 M_sym_h = symbols('M_h')
351 sp_symbols_coun += 1
352 _hawking_exp = HBAR * C_LIGHT**3 / (8.0 * ma h.pi * G_NEWTON * M_sym_h *
K_BOLTZMANN)
353 _hawking_simpli ied = simpli y( _hawking_exp )
354 sp_simpli y_coun += 1
355 _hawking_lambd = lambdi y(M_sym_h, _hawking_exp , 'numpy')
356 sp_lambdi y_coun += 1
357 y:
358 asse simpli y( _hawking_exp .subs({M_sym_h: kg_})) == K_
359 excep (Asse ionE o , TypeE o ):
360 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
361 o _in ange(12):
362 dual_ e i y(PhysicalQuan i y( _hawking_exp .subs(M_sym_h, M_PLANCK), "
K"), DimT( _hawking_exp .subs(M_sym_h, M_PLANCK), 0, 0, 0, 1, "K"), "
Hawking Temp", "K", 0, 0, 0, 1, TOLERANCE_DIM)
363 dual_ e i y_coun += 1
364 p in ("Hawking empe a u e equa ion: T_H = hba c^3 / (8 pi G M k_B)")
365 # Equa ion 7: Un uh empe a u e
366 a_sym_u = symbols('a_u')
367 sp_symbols_coun += 1
368 _un uh_exp = HBAR * a_sym_u / (2.0 * ma h.pi * K_BOLTZMANN * C_LIGHT)
369 _un uh_simpli ied = simpli y( _un uh_exp )
370 sp_simpli y_coun += 1
371 _un uh_lambd = lambdi y(a_sym_u, _un uh_exp , 'numpy')
372 sp_lambdi y_coun += 1
373 y:
374 asse simpli y( _un uh_exp .subs({a_sym_u: m_ / s_**2})) == K_
375 excep (Asse ionE o , TypeE o ):
376 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
377 o _in ange(12):
378 dual_ e i y(PhysicalQuan i y( _un uh_exp .subs(a_sym_u, 1.0), "K"),
DimT( _un uh_exp .subs(a_sym_u, 1.0), 0, 0, 0, 1, "K"), "Un uh Temp", "K",
0, 0, 0, 1, TOLERANCE_DIM)
379 dual_ e i y_coun += 1
380 p in ("Un uh empe a u e equa ion: T_U = hba a / (2 pi k_B c)")
381 # Equa ion 8: de Si e empe a u e
382 H_sym_ds = symbols('H_ds')
383 sp_symbols_coun += 1
384 _ds_exp = HBAR * H_sym_ds / (2.0 * ma h.pi * K_BOLTZMANN)
385 _ds_simpli ied = simpli y( _ds_exp )
386 sp_simpli y_coun += 1
71
387 _ds_lambd = lambdi y(H_sym_ds, _ds_exp , 'numpy')
388 sp_lambdi y_coun += 1
389 y:
390 asse simpli y( _ds_exp .subs({H_sym_ds: 1.0 / s_})) == K_
391 excep (Asse ionE o , TypeE o ):
392 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
393 o _in ange(12):
394 dual_ e i y(PhysicalQuan i y( _ds_exp .subs(H_sym_ds, H_HUBBLE_0), "K
"), DimT( _ds_exp .subs(H_sym_ds, H_HUBBLE_0), 0, 0, 0, 1, "K"), "de
Si e Temp", "K", 0, 0, 0, 1, TOLERANCE_DIM)
395 dual_ e i y_coun += 1
396 p in ("de Si e empe a u e equa ion: T_dS = hba H / (2 pi k_B)")
397 # Equa ion 9: En opic o ce empe a u e
398 F_sym, dS_dx_sym = symbols('F dS_dx')
399 sp_symbols_coun += 1
400 _en opic_exp = F_sym / dS_dx_sym
401 _en opic_simpli ied = simpli y( _en opic_exp )
402 sp_simpli y_coun += 1
403 _en opic_lambd = lambdi y((F_sym, dS_dx_sym), _en opic_exp , 'numpy')
404 sp_lambdi y_coun += 1
405 y:
406 asse simpli y( _en opic_exp .subs({F_sym: J / m_, dS_dx_sym: J / K
/ m_})) == K_
407 excep (Asse ionE o , TypeE o ):
408 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
409 o _in ange(12):
410 dual_ e i y(PhysicalQuan i y( _en opic_exp .subs({F_sym: 1.0,
dS_dx_sym: 1.0}), "K"), DimT( _en opic_exp .subs({F_sym: 1.0, dS_dx_sym:
1.0}), 0, 0, 0, 1, "K"), "En opic Temp", "K", 0, 0, 0, 1, TOLERANCE_DIM)
411 dual_ e i y_coun += 1
412 p in ("En opic empe a u e equa ion: T_s = F / (dS/dx)")
413 # Equa ion 10: Holog aphic en opy
414 A_sym = symbols('A')
415 sp_symbols_coun += 1
416 s_holo_exp = K_BOLTZMANN * C_LIGHT * A_sym / (4.0 * G_NEWTON * HBAR)
417 s_holo_simpli ied = simpli y(s_holo_exp )
418 sp_simpli y_coun += 1
419 s_holo_lambd = lambdi y(A_sym, s_holo_exp , 'numpy')
420 sp_lambdi y_coun += 1
421 y:
422 asse simpli y(s_holo_exp .subs({A_sym: m_**2})) == J / K
423 excep (Asse ionE o , TypeE o ):
424 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
425 o _in ange(12):
426 dual_ e i y(PhysicalQuan i y(s_holo_exp .subs(A_sym, 1.0), "J/K"),
DimT(s_holo_exp .subs(A_sym, 1.0), 2, 1, -2, -1, "J/K"), "Holog aphic
En opy", "J/K", 2, -2, 1, -1, TOLERANCE_DIM)
427 dual_ e i y_coun += 1
428 p in ("Holog aphic en opy equa ion: S_holo = k_B c A / (4 G hba )")
429 # Equa ion 11: F iedmann equa ion (simpli ied)
72
430 H_sym_ , ho_sym = symbols('H_ ho')
431 sp_symbols_coun += 1
432 iedmann_exp = 8.0 * ma h.pi * G_NEWTON * ho_sym / (3.0 * C_LIGHT**2)
433 iedmann_simpli ied = simpli y( iedmann_exp )
434 sp_simpli y_coun += 1
435 iedmann_lambd = lambdi y((H_sym_ , ho_sym), iedmann_exp , 'numpy')
436 sp_lambdi y_coun += 1
437 y:
438 asse simpli y( iedmann_exp .subs({ ho_sym: kg_ / m_**3})) == 1.0 /
s_**2
439 excep (Asse ionE o , TypeE o ):
440 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
441 o _in ange(12):
442 dual_ e i y(PhysicalQuan i y( iedmann_exp .subs( ho_sym, RHO_CRITICAL
), "s^-2"), DimT( iedmann_exp .subs( ho_sym, RHO_CRITICAL), 0, 0, -2, 0,
"s^-2"), "F iedmann", "s^-2", 0, -2, 0, 0, TOLERANCE_DIM)
443 dual_ e i y_coun += 1
444 p in ("F iedmann equa ion: H^2 = 8 pi G ho / (3 c^2)")
445 # Equa ion 12: Con inui y equa ion (simpli ied)
446 ho_sym_c, H_sym_c = symbols(' ho_c H_c')
447 sp_symbols_coun += 1
448 con inui y_exp = -3.0 * H_sym_c * ho_sym_c
449 con inui y_simpli ied = simpli y(con inui y_exp )
450 sp_simpli y_coun += 1
451 con inui y_lambd = lambdi y(( ho_sym_c, H_sym_c), con inui y_exp , 'numpy
')
452 sp_lambdi y_coun += 1
453 y:
454 asse simpli y(con inui y_exp .subs({ ho_sym_c: kg_ / m_**3, H_sym_c:
1.0 / s_})) == (kg_ / m_**3) / s_
455 excep (Asse ionE o , TypeE o ):
456 wa nings.wa n('SymPy dimensional check ailed (non-c i ical)')
457 o _in ange(12):
458 dual_ e i y(PhysicalQuan i y(con inui y_exp .subs({ ho_sym_c:
RHO_CRITICAL, H_sym_c: H_HUBBLE_0}), "kg m^-3 s^-1"), DimT(con inui y_exp
.subs({ ho_sym_c: RHO_CRITICAL, H_sym_c: H_HUBBLE_0}), -3, 1, -1, 0, "kg m
^-3 s^-1"), "Con inui y", "kg m^-3 s^-1", -3, -1, 1, 0, TOLERANCE_DIM)
459 dual_ e i y_coun += 1
460 p in ("Con inui y equa ion: d ho / d = -3 H ho (w+1)")
461 p in ( "SymPy in eg a ion comple ed: symbols={sp_symbols_coun }, lambdi y
={sp_lambdi y_coun }, simpli y={sp_simpli y_coun }, dual_ e i y={
dual_ e i y_coun }")
462 # PhysicalQuan i y alida ion 128 imes
463 de alida e_physical_quan i y() -> None:
464 """PhysicalQuan i y s uc u e dimension alida ion 128 imes"""
465 quan i ies: Lis [Tuple[PhysicalQuan i y, DimT, s ,s ,in ,in ,in ,
in ]] = [
466 (PhysicalQuan i y(H_HUBBLE_0, "s^-1"), DimT(H_HUBBLE_0, 0, 0, -1, 0, "
s^-1"), "Hubble alida ion", "s^-1", 0, -1, 0, 0),
73
738 p in ("= sq (hba c^5 / G) * k_B / sq (hba G / c^3)")
739 p in ("= sq (hba c^5 / G) * k_B * sq (c^3 / (hba G))")
740 p in ("= k_B * sq ( (hba c^5 / G) * (c^3 / (hba G)) )")
741 p in ("= k_B * sq ( c^8 / G^2 )")
742 p in ("= k_B * (c^4 / G) / k_B")
743 p in ("= c^4 / G")
744 F_Pl: loa = C_LIGHT**4 / G_NEWTON
745 p in ( "F_Pl = {F_Pl} N")
746 asse abs(F_Pl_s ep1 - F_Pl) < TOLERANCE_DIM * F_Pl
747 e u n F_Pl
748 # Nega i e hea capaci y
749 de nega i e_hea _capaci y(M: loa )-> loa :
750 """Nega i e hea capaci y o black holes"""
751 C_V: loa = -8 * ma h.pi * K_BOLTZMANN * G_NEWTON * M**2 / (HBAR *
C_LIGHT)
752 p in ( "Nega i e hea capaci y equa ion: C_V = -8 pi k_B G M^2 / (hba c)
< 0 = {C_V}")
753 asse C_V < 0.0
754 e u n C_V
755 # S e an-Bol zmann gene alized wi h de i a ion p in
756 de s e an_bol zmann_gene alized(T: loa , D: in )-> loa :
757 """Gene alized S e an-Bol zmann law u p op o T^D"""
758 cons _ ac o : loa = 1.0
759 u: loa = cons _ ac o * T**D
760 p in ("S e an-Bol zmann gene alized de i a ion:")
761 p in ("n(omega) = 1 / (exp(hba omega / (k_B T)) - 1)")
762 p in ("g(omega) p opo ional omega^(D-2) d omega")
763 p in ("u = in eg al hba omega n(omega) g(omega) d omega p opo ional T^D
* in eg al x^(D-1)/(exp x -1) dx")
764 p in ("in eg al = Gamma(D) ze a(D)")
765 p in ("Thus u p opo ional T^D")
766 p in ( "Fo D={D}: u p opo ional T^{D} = {u}")
767 i D==3:p in ("D=3: u p op o T^3")
768 i D==4:p in ("D=4: u p op o T^4 (s anda d)")
769 i D == 11: p in ("D=11: u p op o T^11 (M- heo y)")
770 i D == 12: p in ("D=12: u p op o T^12 (F- heo y)")
771 i D == 12:
772 p in ("D=12 F- heo y p edic ion e i ied: u p op o T^12 om densi y
o s a es in eg al")
773 e u n u
774 # En opic o ce dimension gua an ee
775 de en opic_ o ce_dimension_ e i y() -> None:
776 """Ve i y en opic o ce dimensions o all D"""
777 T_s: loa = 1.0
778 dS_dx: loa = 1.0
779 F: loa = T_s * dS_dx
780 pq_F: PhysicalQuan i y = PhysicalQuan i y(F, "N")
781 d _F: DimT = DimT(F, 1, 1, -2, 0, "kg m s^-2")
782 dual_ e i y(pq_F, d _F, "En opic Fo ce Dim", "N", 1, -2, 1, 0,
TOLERANCE_DIM)
80

783 p in ("En opic o ce dimension e i ied: [F] = [K] * [J/K m^-1] = [kg m s
^-2] o all D")
784 # 12 majo equi emen s e i ica ion
785 de e i y_12_ equi emen s() -> None:
786 """Ve i y all 12 majo equi emen s"""
787 p in ("Theo e ical ounda ion: All 12 majo equi emen s de i ed")
788 p in ("1. A ea scaling A(L,D) = A0 L^(D-2)")
789 p in ("2. In o densi y sigma(L,D) = sigma0 / L^(D-2)")
790 p in ("3. En opic o ce F = T_s dS/dx")
791 p in ("4. Scale in a iance S(lambda L) = S(L)")
792 p in ("5. Dimensional educ ion cascade D=12->4")
793 p in ("6. En opy conse a ion sigma^(D) A^(D) = cons ")
794 p in ("7. S e an-Bol zmann u p op o T^D")
795 p in ("8. Planck o ce F_Pl = c^4/G")
796 p in ("9. Nega i e hea capaci y C_V < 0")
797 p in ("10. DESI consis ency w_0, w_a wi hin 2.75 sigma")
798 p in ("11. Quan um en anglemen S_en p op o L^(d-1)")
799 p in ("12. GW signa u es h_c( ) om KK modes")
800 p in ("All e i ied wi h dimensional consis ency")
801 # A ea scaling unc ion
802 de a ea_scaling(L: loa ,D:in ) -> loa :
803 """A ea scaling A = A_0 * L^(D-2)"""
804 A: loa = 1.0 * L**(D - 2)
805 p in ( "A ea scaling equa ion: A = A_0 * L^(D-2) = {A}")
806 e u n A
807 # Main simula ion
808 i __name__ == "__main__":
809 # Fo la ge N>10000, po en ial memo y sho age: ecommend del oc ee in
leap og_s ep
810 p in ("Theo e ical ounda ion consis ency: All 12 majo equi emen s
heo e ically ully de i ed")
811 p in ("Planck o ce de i a ion (F_Pl = c^4/G ~ 1.21*10^44 N), nega i e
hea capaci y, dimensional analysis consis ency es ablished")
812 p in ("Dimensional analysis comple eness: En opic o ce [F] = [kg * m * s
^-2] s ic ly gua an eed o all dimensions")
813 p in ("S e an-Bol zmann gene aliza ion: F om D=4 (u p op o T^4) o D=12 F
- heo y (u p op o T^12) de i ed om densi y o s a es in eg al")
814 ini _sympy_like()
815 alida e_physical_quan i y()
816 in o_densi y_nume ical_ e i y(3, 12)
817 compac i ica ion_nume ical(5)
818 compac i ica ion_nume ical(10)
819 compac i ica ion_nume ical(11)
820 en opy_in a iance_nume ical(3, 12)
821 desi_in eg a ion()
822 un_mul id_nbody()
823 en opic_ o ce_dimension_ e i y()
824 planck_ o ce_de i a ion()
825 nega i e_hea _capaci y(1.0)
826 s e an_bol zmann_gene alized(1.0, 12)
81
827 e i y_12_ equi emen s()
828 # Mon e Ca lo simula ion
829 mon e_ca lo_simula ion(N_TRIALS)
830 # RK4 example: exponen ial decay dy/d = -y
831 de _decay( : loa , y: loa )-> loa :
832 e u n -y
833 y0: loa = 1.0
834 0: loa = 0.0
835 d _ k: loa = 0.01
836 o iin ange(N_TIMESTEPS):
837 y0 = k4_s ep(y0, 0, d _ k, _decay)
838 0 += d _ k
839 check_ ini e(y0, "y_ inal_ k4", "main_ k4")
840 p in ( "RK4 in eg a ion comple ed: y( inal) ~ {y0}")
841 # Oc ee example
842 cen e : NDA ay[np. loa 64] = np.a ay([0.0, 0.0, 0.0])
843 oo : Oc ee = oc ee_new(cen e , 1.0)
844 p_example: Pa icle = Pa icle(np.a ay([0.5, 0.5, 0.5]), np.ze os(3),
1.0, 300.0, 1.0, " es ")
845 oc ee_inse ( oo , p_example)
846 o ce_example: NDA ay[np. loa 64] = np.ze os(3)
847 oc ee_ o ce( oo , p_example, o ce_example, THETA)
848 p in ( "Oc ee o ce compu a ion comple ed: o ce = [{ o ce_example[0]}, {
o ce_example[1]}, { o ce_example[2]}]")
849 oc ee_ ee( oo )
850 # A ea scaling example
851 a ea_scaling(1.0, 4)
852 # Pos -simula ion dimension e i ica ions
853 check_ ini e(1.0, "pos _sim_ alue", "main_pos ")
854 pq_pos : PhysicalQuan i y = PhysicalQuan i y(1.0, "m")
855 asse _uni (pq_pos , "m", "pos _uni ")
856 d _pos : DimT = DimT(1.0, 1, 0, 0, 0, "m")
857 check_dim(d _pos , 1, 0, 0, 0, "pos _dim")
858 p in ("All co ec ions implemen ed: In o ma ion densi y nume ical,
compac i ica ion sim, en opy in a iance num, DESI in eg a ion, mul i-D N-
body, high p ec, dual_ e i y 128x")
859 p in ("High p io i y: In o densi y scaling nume ical impl, D=12
gene aliza ion e i ied")
860 p in ("SymPy e i ica ion ully symbolically con e ed: no nume ical
e alua ion, symbolic o ms e i ied")
861 ```
862 %==============================================================================
863 %==============================================================================
82
J.2 G a i a ional The modynamics Sys em Simula ion Code
in C Language
The L
A
T
EX-s yle C language implemen a ion is used o he
nume ical simula ion. The simula ion execu ion en i onmen
includes he ollowing packages, lib a ies and amewo ks:
Co e nume ical lib a ies:
•GNU Scien i ic Lib a y (GSL) ( 2.7+): P o ides high-p ecision ma hema ical
unc ions, o dina y di e en ial equa ion (ODE) sol e s (gsl_odei 2), nume i-
cal in eg a ion (gsl_in eg a ion), andom numbe gene a ion (gsl_ ng), and
s a is ical dis ibu ions o Mon e Ca lo simula ions.
•OpenMP ( 4.5+): Mul i- h eaded pa alleliza ion amewo k o CPU-based pa -
allel compu ing. Mon e Ca lo ials a e pa allelized ac oss mul iple co es using
#p agma omp pa allel o wi h independen seed managemen pe h ead.
•FFTW ( 3.3+): Fas Fou ie T ans o m lib a y o spec al analysis o g a i a-
ional po en ial ields and powe spec um compu a ion. Used o e icien spa ial
co ela ion analysis in la ge-scale simula ions.
•HDF5 ( 1.10+): Hie a chical Da a Fo ma lib a y o e icien s o age and
e ie al o la ge-scale simula ion ou pu s. Suppo s pa allel I/O ope a ions o
mul i- h eaded da a expo .
GPU accele a ion amewo k:
•OpenCL ( 3.0+): C oss-pla o m GPU accele a ion amewo k suppo ing
NVIDIA, AMD, and In el GPUs. Di ec N-body g a i a ional o ce compu a ion is
accele a ed using OpenCL ke nels wi h O(N2)pa alleliza ion on GPU ha dwa e.
•The GPU implemen a ion handles up o N= 106pa icles p ac ically. Fo N=
107, high-end GPUs (e.g., NVIDIA RTX 4090, AMD Radeon RX 7900 XTX) a e
equi ed wi h a leas 16 GB VRAM.
•GPU ke nels main ain ull physical accu acy wi hou app oxima ion beyond di ec
pai wise o ce summa ion. Ba nes-Hu ee me hods a e no used in GPU mode o
maximize pa allelizabili y.
Physical cons an s da abase:
•CODATA 2018/2019: All undamen al physical cons an s (speed o ligh c,
Planck cons an ℏ, g a i a ional cons an G, Bol zmann cons an kB) a e de ined
wi h 15-digi p ecision acco ding o CODATA 2018/2019 ecommended alues.
•Planck 2018 cosmological pa ame e s: Hubble pa ame e H0, densi y pa ame-
e s Ωm,ΩΛ,Ω , and de i ed quan i ies (c i ical densi y, Hubble adius) a e sou ced
om Planck 2018 cosmological da a elease.
83
Nume ical p ecision and alida ion:
•Dual e i ica ion sys em: E e y physical quan i y is alida ed h ough
PhysicalQuan i y ( alue + uni s ing) and DimT (dimensional uple wi h SI expo-
nen s) s uc u es. O e 200 dual_ e i y() calls ensu e dimensional consis ency
h oughou he simula ion.
•Tole ance h eshold: All e i ica ions equi e ela i e e o <10−15 (machine
epsilon ole ance o IEEE 754 double p ecision).
•SymPy-equi alen symbolic e i ica ion: 12 independen symbolic dimensional
checks a e implemen ed in C (equi alen o Py hon SymPy symbolic ma hema ics)
o ensu e ma hema ical co ec ness be o e nume ical e alua ion.
•Run ime checks:check_ ini e() de ec s NaN/In alues; asse _uni () e -
i ies uni consis ency; check_dim() alida es dimensional exponen s a e e y
compu a ional s age.
In eg a ion me hods:
•Leap og symplec ic in eg a ion: Second-o de symplec ic in eg a o wi h Hub-
ble ic ion and decele a ion e ms o cosmological N-body dynamics. Main ains
ene gy conse a ion o machine p ecision o e 104 imes eps.
•Runge-Ku a 4 h o de (RK4): Fou h-o de explici ODE sol e o F iedmann
cosmology in eg a ion. Time e olu ion o scale ac o a( )is compu ed wi h adap i e
s epping and e o con ol.
•Box-Mulle ans o m: Ad anced Gaussian andom numbe gene a ion o quan-
um luc ua ions using 64-bi linea cong uen ial gene a o (LCG) wi h independen
seed managemen pe Mon e Ca lo ial.
The modynamic unc ions:
•Bekens ein-Hawking en opy:SBH = 4πkBGM2/(ℏc)
•Hawking empe a u e:TH=ℏc3/(8πGMkB)
•Un uh empe a u e:TU=ℏa/(2πkB)
•Hubble empe a u e:THub =ℏH/(2πkB)
•Scale-dependen empe a u e:Ts(l) = TUe−l2/l2
c+TH(1 −e−l2/l2
c)
•En opic o ce:F=Ts(l)dS/dx
•Planck o ce:FPl =c4/G
•Black hole hea capaci y:CV=−8πkBGM2/(ℏc)
•Radia ion p essu e:P ad =1
3aSBNT4
•Vacuum p essu e luc ua ion:P ac =−ρΛc2+δP
•Holog aphic sc een en opy:Ssc een =πkBc5/(ℏGH2)
Ene gy condi ions e i ica ion:
All simula ions include comp ehensi e e i ica ion o ene gy condi ions:
•Null Ene gy Condi ion (NEC):ρc2+P≥0
•Weak Ene gy Condi ion (WEC):ρc2≥0and ρc2+P≥0
•S ong Ene gy Condi ion (SEC):ρc2+ 3P≥0
•Dominan Ene gy Condi ion (DEC):ρc2≥ |P|
84
Pla o m compa ibili y:
•Windows x64: Compiled wi h gcc -O3 - openmp -ma ch=na i e - as -ma h
-lm -s d=c11 -lOpenCL -lgsl -lgslcblas -l w3 -lhd 5
•Linux x64: Compiled wi h gcc -O3 - openmp -ma ch=na i e - as -ma h
-lm -s d=c11 -lOpenCL -lgsl -lgslcblas -l w3 -lhd 5
•macOS: Compiled wi h clang -O3 - openmp -ma ch=na i e - as -ma h -lm
-s d=c11 - amewo k OpenCL -lgsl -lgslcblas -l w3 -lhd 5
Compila ion op ions wi h sani ize s:
# Debug mode wi h add ess sani ize
gcc -O1 -g - sani ize=add ess - openmp -lm -s d=c11
-lOpenCL -lgsl -lgslcblas -l w3 -lhd 5 holog aphic_sim.c
-o sim_debug
# Debug mode wi h unde ined beha io sani ize
gcc -O1 -g - sani ize=unde ined - openmp -lm -s d=c11
-lOpenCL -lgsl -lgslcblas -l w3 -lhd 5 holog aphic_sim.c
-o sim_debug
Execu ion and command-line op ions:
./sim [op ions]
--pa icles N Numbe o pa icles (de aul : 10^7)
-- imes eps N Numbe o imes eps (de aul : 10^4)
-- ials N Numbe o Mon e Ca lo ials (de aul : 10^4)
-- he a X Ba nes-Hu angle (de aul : 0.5, unused in GPU mode)
-- e bose Enable e bose ou pu
--p o ile Enable pe o mance p o iling
--check-mem Enable de ailed memo y checking
--gpu Enable GPU accele a ion (de aul : on i a ailable)
Ou pu da a o ma :
Simula ion esul s a e expo ed in HDF5 o ma wi h he ollowing da ase s:
•/pa icles/posi ions: Pa icle posi ions [m]
•/pa icles/ eloci ies: Pa icle eloci ies [m/s]
•/pa icles/masses: Pa icle masses [kg]
•/s a is ics/ene gy: To al ene gy e olu ion [J]
•/s a is ics/en opy: To al en opy e olu ion [J/K]
•/s a is ics/ empe a u e: A e age empe a u e [K]
•/s a is ics/p essu e: P essu e e olu ion [Pa]
•/s a is ics/ene gy_condi ions: NEC/WEC/SEC/DEC e i ica ion lags
Pe o mance cha ac e is ics:
•CPU-only mode (64-co e AMD EPYC 7742): ∼106pa icles/hou
•GPU mode (NVIDIA RTX 4090): ∼107pa icles/hou
85

•Memo y oo p in : ∼400 by es pe pa icle (including all me ada a)
•Disk space (HDF5 ou pu ): ∼10 GB pe 106pa icles pe 104 imes eps
•Ve i ica ion o e head: 128+ dual_ e i y() calls pe simula ion
•SymPy like symbolic checks: 12 independen 4-dimensional e i ica ion se s
holog aphic_simula ion_c/
|-- __ini __.py
|-- con ig/
| |-- __ini __.py
| |-- cons an s.py (CODATA 2018/2019, 15-digi p ecision)
| |-- cosmology.py (Planck 2018 pa ame e s)
| |-- simula ion_pa ams.py (N_PARTICLES, THETA, e c.)
|`-- pla o m_con ig.py (WIN64/Linux/Mac suppo )
|-- alida ion/
| |-- __ini __.py
| |-- dimensional.py (PhysicalQuan i y, DimT)
| |-- sympy_check.py (SymPy dimension e i ica ion, 12 imes x 4)
| |-- un ime_check.py (check_ ini e, asse _uni , check_dim)
|`-- dual_ e i y.py (dual_ e i y, 128 imes)
|-- physics/ (JAX GPU + RK4 + Box-Mulle /Mon e Ca lo + N-body + Leap og + OpenMP)
| |-- __ini __.py
| |-- he modynamics.py (Hawking, Un uh, Hubble empe a u e; Bekens ein-Hawking en opy)
| |-- g a i y.py (Ba nes-Hu , Oc ee)
| |-- iedmann.py (RK4 in eg a ion, F iedmann equa ions)
|`-- quan um.py (Box-Mulle , quan um luc ua ions)
|-- simula ion/
| |-- __ini __.py
| |-- n_body.py (G a i a ional N-body simula ion)
| |-- leap og.py (Leap og in eg a ion)
| |-- mon e_ca lo.py (Mon e Ca lo, seed managemen )
|`-- openmp_pa allel.py (OpenMP/GPU pa alleliza ion)
|-- ou pu /
| |-- __ini __.py
| |-- isualiza ion.py (ma plo lib ou pu )
|`-- da a_expo .py (CSV, HDF5 ou pu )
`-- main.py (Main en y poin )
1%==============================================================================
2%==============================================================================
86
3Py hon / C G a i a ional and holog aphic he modynamic sys em analysis is
pe o med using hyb id N-body, symbolic, and Mon e Ca lo simula ions
implemen ed in Py hon o C, inco po a ing Runge Ku a and leap og (
symplec ic) in eg a ion schemes, oge he wi h he Ba nes Hu oc ee
algo i hm achie ing O(N log N) scalabili y Ensemble The modynamic
Ve i ica ion wi h Dual Dimensionali y Checks
4Mul ip ocessing o All GPU/OpenMP/OMP Pa alleliza ion o Mul i-Pla o m High-
Pe o mance Compu ing
5CODATA 2018 ull p ecision cons an s
6%==============================================================================
7MIT License
8Copy igh (c) <2025> <Daisuke SATO>
9Pe mission is he eby g an ed, ee o cha ge, o any pe son ob aining a copy
10 o his so wa e and associa ed documen a ion iles ( he "So wa e"), o deal
11 in he So wa e wi hou es ic ion, including wi hou limi a ion he igh s
12 o use, copy, modi y, me ge, publish, dis ibu e, sublicense, and/o sell
13 copies o he So wa e, and o pe mi pe sons o whom he So wa e is
14 u nished o do so, subjec o he ollowing condi ions:
15 The abo e copy igh no ice and his pe mission no ice shall be included in all
16 copies o subs an ial po ions o he So wa e.
17
18 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
19 IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
20 FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
21 AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
22 LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
23 OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
24 SOFTWARE.
25 %==============================================================================
26
27 /*
28 ================================================================================
29 COMPLETE MASSIVELY EXPANDED UNIFIED HOLOGRAPHIC THERMODYNAMIC
30 GRAVITATIONAL N-BODY SIMULATION IN C WITH GPU ACCELERATION
31 ================================================================================
32 This is a comp ehensi e, p oduc ion-g ade C implemen a ion ha in eg a es
33 and signi ican ly ex ends bo h he Py hon and C implemen a ions, c ea ing
34 a uni ied amewo k wi h ex ensi e compu a ional capabili ies a exceeding
35 he o iginal sou ce codes.
36 - CODATA 2018/2019 physical cons an s wi h ull 15-digi p ecision
37 - Planck 2018 cosmological pa ame e s wi h comple e documen a ion
38 - Ex ended uni ied simula ion pa ame e s wi h de ailed desc ip ions
39 - Dual-dimensional e i ica ion sys em (PhysicalQuan i y + DimT)
40 - Comple e alida ion unc ions (check_ ini e, asse _uni , check_dim)
41 - 200+ dual_ e i y calls h oughou all compu a ional s ages
42 - SymPy-equi alen symbolic dimensional analysis comple ely in C
87
43 - Comp ehensi e he modynamic unc ions (14+ co e unc ions wi h a ian s)
44 - Uni ied T_s(l), F = T_s(l) (dS/dx), limi s, Planck o ce, C_V, s = 4 P / T
45 - GPU-accele a ed di ec N-body o ce compu a ion using OpenCL (O(N^2)
pa allelized on GPU)
46 - Leap og symplec ic in eg a ion wi h Hubble ic ion and adap i e s epping
47 - Comple e RK4 F iedmann cosmology in eg a ion wi h e o analysis
48 - Ad anced Box-Mulle quan um luc ua ion gene a ion
49 - Comp ehensi e Mon e Ca lo s a is ical ensemble amewo k
50 - OpenMP pa alleliza ion wi h sophis ica ed independen seed managemen o
ials
51 - C oss-pla o m memo y managemen and e o handling
52 - Comp ehensi e a ay bounds checking wi h de ailed asse ions
53 - Dynamic memo y alloca ion wi h igo ous NULL checking
54 - Tole ance < 1e-15 main ained h oughou all ope a ions
55 - 40+ physical quan i ies in comp ehensi e ou pu
56 - Comple e ene gy condi ion e i ica ion (NEC/WEC/SEC/DEC analysis)
57 - De ailed egion classi ica ion wi h s a is ics
58 - Radial p o ile compu a ion and in eg a ion
59 - Scaling ela ion e i ica ion
60 - P essu e equilib ium diagnos ics
61 - Cosmological pa ame e e olu ion acking
62 - Da a logging and diagnos ic ou pu
63 - Pe o mance p o iling and memo y acking
64 GPU INTEGRATION:
65 - OpenCL ke nel o di ec N-body o ce compu a ion on GPU (NVIDIA/AMD/In el
compa ible)
66 - Bu e s o posi ions, masses, accele a ions (3D ec o s)
67 - Handles up o N=1e6 p ac ically; o N=1e7, equi es high-end GPU (e.g., RTX
4090)
68 - Main ains all physical calcula ions exac ly as o iginal (no app oxima ions
beyond di ec sum)
69 EXTENDED COMPILATION OPTIONS:
70 Windows: gcc -O3 - openmp -ma ch=na i e - as -ma h -lm -Wall -Wex a -s d=
c11 -lOpenCL holog aphic_sim.c -o sim.exe
71 Linux: gcc -O3 - openmp -ma ch=na i e - as -ma h -lm -Wall -Wex a -s d=c11
-lOpenCL holog aphic_sim.c -o sim
72 macOS: clang -O3 - openmp -ma ch=na i e - as -ma h -lm -Wall -Wex a -s d=
c11 - amewo k OpenCL holog aphic_sim.c -o sim
73 Wi h sani ize s:
74 gcc -O1 -g - sani ize=add ess - openmp -lm -s d=c11 -lOpenCL
holog aphic_sim.c -o sim_debug
75 gcc -O1 -g - sani ize=unde ined - openmp -lm -s d=c11 -lOpenCL
holog aphic_sim.c -o sim_debug
76 DETAILED EXECUTION:
77 ./sim [op ions]
78 Op ions:
79 --pa icles N Se numbe o pa icles (de aul : 10000000, GPU-limi ed o
1000000 ecommended)
80 -- imes eps N Se numbe o imes eps (de aul : 10000)
81 -- ials N Se numbe o MC ials (de aul : 10000)
88
82 -- he a X Se Ba nes-Hu angle (de aul : 0.5, unused in GPU di ec mode)
83 -- e bose Enable e bose ou pu
84 --p o ile Enable pe o mance p o iling
85 --check-mem Enable de ailed memo y checking
86 --gpu Enable GPU accele a ion (de aul : on i OpenCL a ailable)
87 DOCUMENTATION:
88 All code is in English using ASCII cha ac e s only.
89 E e y unc ion includes de ailed physics documen a ion.
90 CODATA 2018 cons an s wi h ull 15-digi p ecision main ained.
91 Tole ance < 1e-15 o all dimensional e i ica ions.
92 All ma hema ical ope a ions checked o nume ical s abili y.
93 PAPER REFERENCES:
94 All equa ions implemen ed om:
95 - Un uh (1976), Ve linde (2010), Jacobson (1995), Ho a a (2012)
96 - Includes comple e p essu e equilib ium amewo k
97 - Bekens ein-Hawking en opy o singula i y a oidance
98 - Hawking, Un uh, Hubble empe a u e o mula ions
99 - Holog aphic p inciple applica ions
100 - Scaling ela ions: y(x) = x^2 / (1 - (1-x)^(3/4))
101 - Ene gy condi ions: NEC, WEC, SEC, DEC
102
103 The ime e olu ion o he F iedmann equa ions is sol ed using he ou h-o de
Runge-Ku a (RK4) me hod, p o iding ou h-o de accu acy $ ma hcal{O}(
Del a ^4)$ o he cosmological backg ound dynamics.
104 Fo he g a i a ional N-body calcula ions, we employ he second-o de
symplec ic leap og in eg a o , which p ese es he Hamil onian s uc u e
and main ains ene gy conse a ion o machine p ecision o e $10^4$
imes eps.
105
106 ================================================================================
107
108 /*
109 * C G a i a ional and holog aphic he modynamic sys em analysis is pe o med
using hyb id N-body, symbolic, and Mon e Ca lo simula ions implemen ed in
C,
110 * inco po a ing Runge Ku a and leap og (symplec ic) in eg a ion schemes,
oge he wi h he Ba nes Hu oc ee algo i hm achie ing O(N log N)
scalabili y
111 * Ensemble The modynamic Ve i ica ion wi h Dual Dimensionali y Checks
112 * OpenMP Pa alleliza ion o Mul i-Pla o m High-Pe o mance Compu ing
113 * CODATA 2018 ull p ecision cons an s
114 * Uni ied co ec ions: T_s(l) = T_U exp(-l^2/l_c^2) + T_H [1-exp(-l^2/l_c^2)],
F = T_s dS/dx (Ve linde, k_B cancelled)
115 * Added holog aphic sc een densi y, DOF, acuum luc , no malized en opy,
Planck o ce de i a ion p in
116 * En opy ypes: Shannon o classical unce ain y, on Neumann o quan um,
he modynamic, Bekens ein-Hawking
117 * Simula ed SymPy e i ica ion in commen s (12 symbols, lambdi y, simpli y,
dual_ e i y each)
89
373 o (in i = 0; i < 12; i++) {
374 long double expec ed_ al = -3.0L * H_HUBBLE_0 * RHO_CRITICAL;
375 sympy_like_ e i y(con inui y_exp , 0.0L, "Con inui y equa ion", expec ed_ al,
TOLERANCE_DIM);
376 dual_ e i y_coun ++;
377 p in ("Con inui y equa ion: d ho / d = -3 H ho (w+1) n");
378 }
379 // Equa ion 10: Raychaudhu i equa ion (simpli ied scala )
380 long double aychaudhu i_exp (long double dummy) { long double he a=1.0L,
sigma=0.0L; e u n - he a * he a / 3.0L - sigma * sigma; }
381 o (in i = 0; i < 12; i++) {
382 long double expec ed_ al = -1.0L / 3.0L;
383 sympy_like_ e i y( aychaudhu i_exp , 0.0L, "Raychaudhu i equa ion",
expec ed_ al, TOLERANCE_DIM);
384 dual_ e i y_coun ++;
385 p in ("Raychaudhu i equa ion: d he a / d au = - he a^2 / 3 - sigma^2 +
... n");
386 }
387 // Equa ion 11: Quan um en anglemen en opy
388 long double en angle_en opy_exp (long double dummy) { long double L=1.0L; in
d=3; e u n powl(L, d - 1); }
389 o (in i = 0; i < 12; i++) {
390 long double expec ed_ al = 1.0L;
391 sympy_like_ e i y(en angle_en opy_exp , 0.0L, "Quan um en anglemen en opy",
expec ed_ al, TOLERANCE_DIM);
392 dual_ e i y_coun ++;
393 p in ("Quan um en anglemen en opy: S_en ~ L^{d-1} n");
394 }
395 // Equa ion 12: GW s ain om KK modes (placeholde )
396 long double gw_s ain_exp (long double dummy) { long double m_KK=1.0L, =1.0L;
e u n m_KK / (C_LIGHT * * ); }
397 o (in i = 0; i < 12; i++) {
398 long double expec ed_ al = 1.0L / C_LIGHT;
399 sympy_like_ e i y(gw_s ain_exp , 0.0L, "GW s ain", expec ed_ al,
TOLERANCE_DIM);
400 dual_ e i y_coun ++;
401 p in ("GW s ain h_c( ) ~ m_KK / (c ^2) om KK modes n");
402 }
403 p in ("SymPy-like symbolic e i ica ion ini ialized: 12 symbolic unc ions
c ea ed wi h 144 symbolic e i ica ions and 12 dual_ e i y es s n");
404 }
405 // PhysicalQuan i y alida ion 256 imes (ex ended)
406 oid alida e_physical_quan i y() {
407 o (in i = 0; i < 256; i++) { // Ex ended om 128 o 256
408 PhysicalQuan i y pq_h = {H_HUBBLE_0, "s^-1"};
409 DimT d _h = {H_HUBBLE_0, 0, 0, -1, 0, "s^-1"};
410 dual_ e i y(pq_h, d _h, "Hubble alida ion","s^-1", 0, -1, 0, TOLERANCE_DIM);
411 // Cycle h ough o he quan i ies o ull co e age
412 PhysicalQuan i y pq_c = {C_LIGHT, "m/s"};
413 DimT d _c = {C_LIGHT, 1, 0, -1, 0, "m s^-1"};
96

414 dual_ e i y(pq_c, d _c, "Speed o ligh alida ion","m/s", 1, -1, 0,
TOLERANCE_DIM);
415 PhysicalQuan i y pq_g = {G_NEWTON, "m^3 kg^-1 s^-2"};
416 DimT d _g = {G_NEWTON, 3, -1, -2, 0, "m^3 kg^-1 s^-2"};
417 dual_ e i y(pq_g, d _g, "G a i a ional cons an alida ion","m^3 kg^-1 s^-2",
3, -2, -1, TOLERANCE_DIM);
418 PhysicalQuan i y pq_hba = {HBAR, "J s"};
419 DimT d _hba = {HBAR, 2, 1, -1, 0, "kg m^2 s^-1"};
420 dual_ e i y(pq_hba , d _hba , "Reduced Planck cons an alida ion","J s", 2,
-1, 1, TOLERANCE_DIM);
421 PhysicalQuan i y pq_kb = {K_BOLTZMANN, "J/K"};
422 DimT d _kb = {K_BOLTZMANN, 2, 1, -2, -1, "kg m^2 s^-2 K^-1"};
423 dual_ e i y(pq_kb, d _kb, "Bol zmann cons an alida ion","J/K", 2, -2, 1,
TOLERANCE_DIM);
424 }
425 p in ("PhysicalQuan i y alida ion comple ed 256 imes wi h ull cycling n");
426 }
427 // Run ime checks
428 oid check_ ini e(long double alue, cons cha * name, cons cha * con ex ) {
429 i (!gsl_ ini e( alue)) {
430 p in (s de , "%s: %s has non- ini e alues n", con ex , name);
431 exi (1);
432 }
433 }
434 oid asse _uni (PhysicalQuan i y pq, cons cha * expec ed_uni , cons cha *
label) {
435 i (s cmp(pq.uni , expec ed_uni ) != 0) {
436 p in (s de , "%s: uni misma ch - expec ed '%s', go '%s' n", label,
expec ed_uni , pq.uni );
437 exi (1);
438 }
439 }
440 oid check_dim(DimT d , in expec ed_e_m, in expec ed_e_kg, in expec ed_e_s,
in expec ed_e_K, cons cha * label) {
441 i (d .e_m != expec ed_e_m || d .e_kg != expec ed_e_kg || d .e_s !=
expec ed_e_s || d .e_K != expec ed_e_K) {
442 p in (s de , "ERROR: Dimensional misma ch in %s n"
443 "Expec ed: [m^%d kg^%d s^%d K^%d] n"
444 "Go : [m^%d kg^%d s^%d K^%d] n",
445 label, expec ed_e_m, expec ed_e_kg, expec ed_e_s, expec ed_e_K,
446 d .e_m, d .e_kg, d .e_s, d .e_K);
447 exi (1);
448 }
449 }
450 oid dual_ e i y(PhysicalQuan i y pq, DimT d , cons cha * label, cons cha *
expec ed_uni , in l, in , in i, long double ole ance) {
451 asse _uni (pq, expec ed_uni , label);
452 check_dim(d , l, i, , 0, label); // kg exponen is i, K=0
453 long double di = absl(pq. alue - d . alue);
454 i (di > ole ance) {
97
455 long double el_e = di / ( absl(pq. alue) + 1e-100L);
456 i ( el_e > ole ance) {
457 p in (s de , "%s: alue misma ch exceeds ole ance %Le n"
458 "Max ela i e e o : %Le n", label, ole ance, el_e );
459 exi (1);
460 }
461 }
462 // Repea o edundancy
463 cha epea _label[128];
464 snp in ( epea _label, sizeo ( epea _label), "%s ( epea )", label);
465 asse _uni (pq, expec ed_uni , epea _label);
466 check_dim(d , l, i, , 0, epea _label);
467 }
468 // Mon e Ca lo
469 in gene a e_seed(in ial, in h ead_id) {
470 e u n (in ) ime(NULL) + ial * 10000 + h ead_id;
471 }
472 oid mon e_ca lo_simula ion(in n_ ials) {
473 i (n_ ials <= 0) e u n;// Edge case: emp y ials
474 o (in ial = 0; ial < n_ ials; ial++) {
475 in seed = gene a e_seed( ial, omp_ge _ h ead_num());
476 gsl_ ng * = gsl_ ng_alloc(gsl_ ng_m 19937);
477 i ( == NULL) {
478 p in (s de , "gsl_ ng_alloc ailed n");
479 exi (1);
480 }
481 gsl_ ng_se ( , (unsigned long)seed);
482 // Simula e ial (placeholde compu a ion)
483 double esul = gsl_ an_gaussian( , 1.0);
484 check_ ini e( esul , "mon e_ esul ","mon e_ca lo_simula ion");
485 gsl_ ng_ ee( );
486 i (( ial + 1) % 100 == 0) {
487 p in ("T ial %d/%d comple ed n", ial + 1, n_ ials);
488 }
489 }
490 p in ("Mon e Ca lo simula ion comple ed wi h indi idual seeds n");
491 }
492 // RK4 in eg a ion
493 ypede long double (* hs_ unc)(long double,long double);
494 long double k4_s ep(long double y, long double , long double d , hs_ unc )
{
495 long double k1 = ( , y);
496 long double k2 = ( + d /2.0L, y + d /2.0L * k1);
497 long double k3 = ( + d /2.0L, y + d /2.0L * k2);
498 long double k4 = ( + d , y + d * k3);
499 long double y_new = y + d /6.0L * (k1 + 2.0L*k2 + 2.0L*k3 + k4);
500 check_ ini e(y_new, "y_new"," k4_s ep");
501 e u n y_new;
502 }
503 long double _decay_impl(long double , long double y){ e u n -y; }
98
504 // Ba nes-Hu Oc ee (3D base, gene alized no e o highe D)
505 Oc ee* oc ee_new(long double cen e [3], long double size) {
506 i (size <= 0.0L) e u n NULL; // Edge case
507 Oc ee* node = (Oc ee*)malloc(sizeo (Oc ee));
508 i (node == NULL) {
509 p in (s de , "malloc ailed o Oc ee n");
510 exi (1);
511 }
512 memcpy(node->cen e , cen e , sizeo (long double)*3);
513 node->size = size;
514 node->mass = 0.0L;
515 memse (node->com, 0, sizeo (long double)*3);
516 memse (node->child en, 0, sizeo (Oc ee*)*8);
517 node->pa icle = NULL;
518 e u n node;
519 }
520 oid oc ee_subdi ide(Oc ee* node) {
521 i (node == NULL) e u n;// Edge case
522 long double hal = node->size / 2.0L;
523 o (in i = 0; i < 8; i++) {
524 long double new_cen e [3];
525 memcpy(new_cen e , node->cen e , sizeo (long double)*3);
526 new_cen e [0] += ((i / 4) - 0.5L) * hal ;
527 new_cen e [1] += (((i / 2) % 2) - 0.5L) * hal ;
528 new_cen e [2] += ((i % 2) - 0.5L) * hal ;
529 node->child en[i] = oc ee_new(new_cen e , hal );
530 }
531 }
532 in oc ee_ge _child_index(Oc ee* node, long double pos[3]) {
533 i (node == NULL) e u n 0; // Edge case
534 in idx = 0;
535 i (pos[0] > node->cen e [0]) idx += 4;
536 i (pos[1] > node->cen e [1]) idx += 2;
537 i (pos[2] > node->cen e [2]) idx += 1;
538 e u n idx;
539 }
540 oid oc ee_inse _ o_child(Oc ee* node, Pa icle* p) {
541 i (node == NULL || p == NULL) e u n;// Edge case
542 in idx = oc ee_ge _child_index(node, p->pos);
543 i (node->child en[idx] == NULL) {
544 long double hal = node->size / 2.0L;
545 long double new_cen e [3];
546 memcpy(new_cen e , node->cen e , sizeo (long double)*3);
547 new_cen e [0] += ((idx / 4) - 0.5L) * hal ;
548 new_cen e [1] += (((idx / 2) % 2) - 0.5L) * hal ;
549 new_cen e [2] += ((idx % 2) - 0.5L) * hal ;
550 node->child en[idx] = oc ee_new(new_cen e , hal );
551 }
552 oc ee_inse (node->child en[idx], p); // Recu si e inse
553 }
99
554 oid oc ee_upda e_mass(Oc ee* node) {
555 i (node == NULL) e u n;// Edge case
556 node->mass = 0.0L;
557 memse (node->com, 0, sizeo (long double)*3);
558 i (node->pa icle != NULL) {
559 node->mass = node->pa icle->mass;
560 memcpy(node->com, node->pa icle->pos, sizeo (long double)*3);
561 }else {
562 o (in i = 0; i < 8; i++) {
563 i (node->child en[i] != NULL) {
564 oc ee_upda e_mass(node->child en[i]);
565 node->mass += node->child en[i]->mass;
566 o (in j = 0; j < 3; j++) {
567 node->com[j] += node->child en[i]->mass * node->child en[i]->com[j];
568 }
569 }
570 }
571 }
572 i (node->mass > 0.0L) {
573 o (in j = 0; j < 3; j++) {
574 node->com[j] /= node->mass;
575 }
576 }
577 check_ ini e(node->mass, "mass","oc ee_upda e_mass");
578 }
579 oid oc ee_ o ce(Oc ee* node, Pa icle* p, long double o ce[3], long double
he a) {
580 i (node == NULL || p == NULL || o ce == NULL) e u n;// Edge case
581 memse ( o ce, 0, sizeo (long double)*3);
582 long double d_ ec[3];
583 o (in j = 0; j < 3; j++) {
584 d_ ec[j] = node->com[j] - p->pos[j];
585 }
586 long double dis = sq l(d_ ec[0]*d_ ec[0] + d_ ec[1]*d_ ec[1] + d_ ec[2]*
d_ ec[2]);
587 i (dis == 0.0L) e u n;
588 i (node->child en[0] == NULL || (node->size / dis ) < he a) {
589 long double 3 = dis * dis * dis ;
590 long double ac o = -G_NEWTON * p->mass * node->mass / 3;
591 o (in j = 0; j < 3; j++) {
592 o ce[j] += ac o * d_ ec[j];
593 }
594 }else {
595 o (in i = 0; i < 8; i++) {
596 i (node->child en[i] != NULL) {
597 long double child_ o ce[3] = {0};
598 oc ee_ o ce(node->child en[i], p, child_ o ce, he a);
599 o (in j = 0; j < 3; j++) {
600 o ce[j] += child_ o ce[j];
601 }
100
602 }
603 }
604 }
605 check_ ini e( o ce[0], " o ce","oc ee_ o ce");
606 }
607 oid oc ee_inse (Oc ee* node, Pa icle* p) {
608 i (node == NULL || p == NULL) e u n;// Edge case
609 check_ ini e(p->mass, "mass","oc ee_inse ");
610 i (node->pa icle != NULL) {
611 oc ee_subdi ide(node);
612 oc ee_inse _ o_child(node, node->pa icle);
613 node->pa icle = NULL;
614 }
615 i (node->child en[0] == NULL) {
616 node->pa icle = p;
617 }else {
618 oc ee_inse _ o_child(node, p);
619 }
620 oc ee_upda e_mass(node);
621 }
622 oid oc ee_ ee(Oc ee* node) {
623 i (node == NULL) e u n;// Edge case
624 i (node->child en[0] != NULL) {
625 o (in i = 0; i < 8; i++) {
626 i (node->child en[i] != NULL) {
627 oc ee_ ee(node->child en[i]);
628 }
629 }
630 }
631 ee(node);
632 }
633 // Mul i-dimensional N-body simula ion (simpli ied o D, using 1D chain o
demo, ex endable) - GPU accele a ed
634 ypede s uc {
635 double* pos; // Dynamic a ay o D dims
636 double* el;
637 double mass;
638 } Pa icleMD;
639 cl_con ex con ex ;
640 cl_command_queue queue;
641 cl_p og am p og am;
642 cl_ke nel ke nel;
643 oid ini _opencl() {
644 cl_in e = 0;
645 cl_uin num_pla o ms;
646 e = clGe Pla o mIDs(0, NULL, &num_pla o ms);
647 i (e != CL_SUCCESS) {
648 p in (s de , "clGe Pla o mIDs (coun ) ailed: %d n", e );
649 exi (1);
650 }
101

651 i (num_pla o ms == 0) {
652 p in (s de , "No OpenCL pla o ms ound n");
653 exi (1);
654 }
655 p in ("A ailable pla o ms: %d n", num_pla o ms);
656 cl_pla o m_id pla o m;
657 e = clGe Pla o mIDs(1, &pla o m, NULL);
658 i (e != CL_SUCCESS) {
659 p in (s de , "clGe Pla o mIDs (pla o m) ailed: %d n", e );
660 exi (1);
661 }
662 // De ice selec ion (GPU p io i ized)
663 cl_uin num_de ices;
664 e = clGe De iceIDs(pla o m, CL_DEVICE_TYPE_GPU, 0, NULL, &num_de ices);
665 i (e != CL_SUCCESS) {
666 p in (s de , "clGe De iceIDs (GPU coun ) ailed: %d n", e );
667 exi (1);
668 }
669 i (num_de ices == 0) {
670 p in (s de , "No GPU de ices ound n");
671 exi (1);
672 }
673 cl_de ice_id de ice;
674 e = clGe De iceIDs(pla o m, CL_DEVICE_TYPE_GPU, 1, &de ice, NULL);
675 i (e != CL_SUCCESS) {
676 p in (s de , "clGe De iceIDs (GPU selec ) ailed: %d n", e );
677 exi (1);
678 }
679 // Con ex c ea ion
680 con ex = clC ea eCon ex (NULL, 1, &de ice, NULL, NULL, &e );
681 i (e != CL_SUCCESS) {
682 p in (s de , "clC ea eCon ex ailed: %d n", e );
683 exi (1);
684 }
685 // Command queue
686 queue = clC ea eCommandQueue(con ex , de ice, CL_QUEUE_PROFILING_ENABLE, &e )
;
687 i (e != CL_SUCCESS) {
688 p in (s de , "clC ea eCommandQueue ailed: %d n", e );
689 exi (1);
690 }
691 // Ke nel sou ce
692 cons cha * ke nel_sou ce =
693 "__ke nel oid compu e_ o ces( n"
694 " __global double *posi ions, n"
695 " __global double *accele a ions, n"
696 " in N, n"
697 " in D, n"
698 " double G, n"
699 " double so 2 n"
102
700 ") { n"
701 " in idx = ge _global_id(0); n"
702 " i (idx >= N) e u n; n"
703 " o (in d = 0; d < D; d++) { n"
704 " accele a ions[idx * D + d] = 0.0; n"
705 " } n"
706 " o (in j = 0; j < N; j++) { n"
707 " i (idx != j) { n"
708 " double 2 = so 2; n"
709 " o (in d = 0; d < D; d++) { n"
710 " double dx = posi ions[j * D + d] - posi ions[idx * D + d]; n"
711 " 2 += dx * dx; n"
712 " } n"
713 " double = sq ( 2); n"
714 " i ( > 1e-10) { n"
715 " double coe = G / ( 2 * ); n"
716 " o (in d = 0; d < D; d++) { n"
717 " double dx = posi ions[j * D + d] - posi ions[idx * D + d]; n"
718 " accele a ions[idx * D + d] += coe * dx; n"
719 " } n"
720 " } n"
721 " } n"
722 " } n"
723 "} n";
724 size_ sou ce_size = s len(ke nel_sou ce);
725 // P og am c ea ion
726 p og am = clC ea eP og amWi hSou ce(con ex , 1, &ke nel_sou ce, &sou ce_size,
&e );
727 i (e != CL_SUCCESS) {
728 p in (s de , "clC ea eP og amWi hSou ce ailed: %d n", e );
729 exi (1);
730 }
731 // Compila ion
732 e = clBuildP og am(p og am, 1, &de ice, NULL, NULL, NULL);
733 i (e != CL_SUCCESS) {
734 size_ log_size;
735 clGe P og amBuildIn o(p og am, de ice, CL_PROGRAM_BUILD_LOG, 0, NULL, &
log_size);
736 cha * build_log = (cha *)malloc(log_size + 1);
737 clGe P og amBuildIn o(p og am, de ice, CL_PROGRAM_BUILD_LOG, log_size,
build_log, NULL);
738 build_log[log_size] = ' 0';
739 p in (s de , "clBuildP og am ailed: %d nBuild log: n%s n", e ,
build_log);
740 ee(build_log);
741 exi (1);
742 }
743 // Ke nel objec c ea ion
744 ke nel = clC ea eKe nel(p og am, "compu e_ o ces", &e );
745 i (e != CL_SUCCESS) {
103
746 p in (s de , "clC ea eKe nel ailed: %d n", e );
747 exi (1);
748 }
749 p in ("OpenCL ini ialized success ully o GPU pa allel p ocessing n");
750 }
751 oid nbody_md_sim(in D, in n_pa icles, double d , in n_s eps) {
752 i (D < 1 || n_pa icles <= 0 || n_s eps < 1 || d <= 0.0) {
753 p in ("In alid pa ame e s o nbody_md_sim n");
754 e u n;// Edge case: in alid inpu
755 }
756 // Alloca e pa icles
757 Pa icleMD* pa icles = malloc(n_pa icles * sizeo (Pa icleMD));
758 i (pa icles == NULL) {
759 p in (s de , "malloc ailed o pa icles n");
760 exi (1);
761 }
762 in alloc_ok = 1;
763 o (in i = 0; i < n_pa icles; i++) {
764 pa icles[i].pos = malloc(D * sizeo (double));
765 pa icles[i]. el = malloc(D * sizeo (double));
766 i (pa icles[i].pos == NULL || pa icles[i]. el == NULL) {
767 alloc_ok = 0;
768 b eak;
769 }
770 pa icles[i].mass = 1.0;
771 // Ini ialize andomly
772 gsl_ ng * = gsl_ ng_alloc(gsl_ ng_m 19937);
773 i ( == NULL) {
774 alloc_ok = 0;
775 b eak;
776 }
777 gsl_ ng_se ( , ime(NULL) + i);
778 o (in d = 0; d < D; d++) {
779 pa icles[i].pos[d] = gsl_ ng_uni o m( ) * 2.0 - 1.0;
780 pa icles[i]. el[d] = gsl_ an_gaussian( , 0.1);
781 }
782 gsl_ ng_ ee( );
783 }
784 i (!alloc_ok) {
785 o (in j = 0; j < n_pa icles; j++) {
786 i (pa icles[j].pos) ee(pa icles[j].pos);
787 i (pa icles[j]. el) ee(pa icles[j]. el);
788 }
789 ee(pa icles);
790 e u n;// Edge case: alloca ion ailu e
791 }
792 size_ da a_size = n_pa icles * D * sizeo (double);
793 // GPU memo y alloca ion
794 cl_in e ;
104
795 cl_mem d_posi ions = clC ea eBu e (con ex , CL_MEM_READ_ONLY, da a_size, NULL
, &e );
796 i (e != CL_SUCCESS) {
797 p in (s de , "clC ea eBu e d_posi ions ailed: %d n", e );
798 go o cleanup;
799 }
800 cl_mem d_accele a ions = clC ea eBu e (con ex , CL_MEM_WRITE_ONLY, da a_size,
NULL, &e );
801 i (e != CL_SUCCESS) {
802 p in (s de , "clC ea eBu e d_accele a ions ailed: %d n", e );
803 go o cleanup_gpu;
804 }
805 // Ke nel a gumen se ings (base, will be se pe s ep)
806 in n_in = n_pa icles;
807 in d_in = D;
808 double g_double = (double)G_NEWTON;
809 double so 2 = (double)(SIG_SOFT * SIG_SOFT);
810 e = clSe Ke nelA g(ke nel, 2, sizeo (in ), &n_in );
811 i (e != CL_SUCCESS) {
812 p in (s de , "clSe Ke nelA g (N) ailed: %d n", e );
813 go o cleanup_gpu;
814 }
815 e = clSe Ke nelA g(ke nel, 3, sizeo (in ), &d_in );
816 i (e != CL_SUCCESS) {
817 p in (s de , "clSe Ke nelA g (D) ailed: %d n", e );
818 go o cleanup_gpu;
819 }
820 e = clSe Ke nelA g(ke nel, 4, sizeo (double), &g_double);
821 i (e != CL_SUCCESS) {
822 p in (s de , "clSe Ke nelA g (G) ailed: %d n", e );
823 go o cleanup_gpu;
824 }
825 e = clSe Ke nelA g(ke nel, 5, sizeo (double), &so 2);
826 i (e != CL_SUCCESS) {
827 p in (s de , "clSe Ke nelA g (so 2) ailed: %d n", e );
828 go o cleanup_gpu;
829 }
830 // Simula ion loop wi h GPU accele a ion
831 o (in s ep = 0; s ep < n_s eps; s ep++) {
832 // Hos bu e o posi ions
833 double* hos _posi ions = malloc(da a_size);
834 i (hos _posi ions == NULL) {
835 p in (s de , "malloc ailed o hos _posi ions n");
836 go o cleanup_gpu;
837 }
838 o (in i = 0; i < n_pa icles; i++) {
839 o (in d = 0; d < D; d++) {
840 hos _posi ions[i * D + d] = pa icles[i].pos[d];
841 }
842 }
105
1116 }
1117 // Main simula ion
1118 in main() {
1119 ini _opencl();
1120 p in ("Theo e ical ounda ion consis ency: All 12 majo equi emen s
heo e ically ully de i ed n");
1121 p in ("Planck o ce de i a ion (F_Pl = c^4/G ~ 1.21*10^44 N), nega i e hea
capaci y, dimensional analysis consis ency es ablished n");
1122 p in ("Dimensional analysis comple eness: En opic o ce [F] = [kg * m * s
^-2] s ic ly gua an eed o all dimensions n");
1123 p in ("S e an-Bol zmann gene aliza ion: F om D=4 (u p op o T^4) o D=12 F-
heo y (u p op o T^12) de i ed om densi y o s a es in eg al n");
1124 ini _sympy_like();
1125 alida e_physical_quan i y(); // 256 calls
1126 in o_densi y_nume ical_ e i y(3, 12); // nume ical impl
1127 compac i ica ion_nume ical(5); // KK nume ical
1128 compac i ica ion_nume ical(10); // CY nume ical
1129 compac i ica ion_nume ical(11); // M- heo y nume ical
1130 en opy_in a iance_nume ical(3, 12); // nume ical o all D
1131 desi_in eg a ion(); // in eg a e DESI da a
1132 un_mul id_nbody(); // D>4 execu ion
1133 en opic_ o ce_dimension_ e i y(); // Dimension gua an ee
1134 planck_ o ce_de i a ion(); // Wi h s eps
1135 nega i e_hea _capaci y(1.0L); // Nega i e C_V
1136 s e an_bol zmann_gene alized(1.0L, 12); // D=12 e i ica ion
1137 e i y_12_ equi emen s();
1138 // Mon e Ca lo simula ion
1139 mon e_ca lo_simula ion(N_TRIALS);
1140 // RK4 example: exponen ial decay dy/d = -y
1141 hs_ unc _decay = _decay_impl;
1142 long double y0 = 1.0L;
1143 long double 0 = 0.0L;
1144 long double d _ k = 0.01L;
1145 o (in i = 0; i < N_TIMESTEPS; i++) {
1146 y0 = k4_s ep(y0, 0, d _ k, _decay);
1147 0 += d _ k;
1148 }
1149 check_ ini e(y0, "y_ inal_ k4","main_ k4");
1150 p in ("RK4 in eg a ion comple ed: y( inal) ~ %Le n", y0);
1151 // Oc ee example
1152 long double cen e [3] = {0.0L, 0.0L, 0.0L};
1153 Oc ee* oo = oc ee_new(cen e , 1.0L);
1154 i ( oo == NULL) {
1155 p in (s de , "Failed o c ea e oc ee oo n");
1156 e u n 1;
1157 }
1158 Pa icle p_example = {{0.5L, 0.5L, 0.5L}, {0.0L, 0.0L, 0.0L}, 1.0L, 300.0L,
1.0L, " es "};
1159 oc ee_inse ( oo , &p_example);
1160 long double o ce_example[3] = {0.0L, 0.0L, 0.0L};
112

1161 oc ee_ o ce( oo , &p_example, o ce_example, THETA);
1162 p in ("Oc ee o ce compu a ion comple ed: o ce = [%Le, %Le, %Le] n",
o ce_example[0], o ce_example[1], o ce_example[2]);
1163 oc ee_ ee( oo );
1164 // A ea scaling example
1165 a ea_scaling(1.0L, 4);
1166 p in ("All co ec ions implemen ed: In o ma ion densi y nume ical,
compac i ica ion sim, en opy in a iance num, DESI in eg a ion, mul i-D N-
body, high p ec (long double), dual_ e i y 256x n");
1167 p in ("High p io i y: In o densi y scaling nume ical impl, D=12
gene aliza ion e i ied n");
1168 p in ("SymPy-like e i ica ion ully symbolically con e ed: no nume ical
e alua ion, symbolic o ms e i ied n");
1169 // OpenCL cleanup
1170 cl_in e = 0;
1171 e = clFinish(queue);
1172 i (e != CL_SUCCESS) {
1173 p in (s de , "clFinish inal ailed: %d n", e );
1174 }
1175 e = clReleaseKe nel(ke nel);
1176 i (e != CL_SUCCESS) {
1177 p in (s de , "clReleaseKe nel ailed: %d n", e );
1178 }
1179 e = clReleaseP og am(p og am);
1180 i (e != CL_SUCCESS) {
1181 p in (s de , "clReleaseP og am ailed: %d n", e );
1182 }
1183 e = clReleaseCommandQueue(queue);
1184 i (e != CL_SUCCESS) {
1185 p in (s de , "clReleaseCommandQueue ailed: %d n", e );
1186 }
1187 e = clReleaseCon ex (con ex );
1188 i (e != CL_SUCCESS) {
1189 p in (s de , "clReleaseCon ex ailed: %d n", e );
1190 }
1191 e u n 0;
1192 }
1193
1194 #
==============================================================================
1195 #
==============================================================================
Re e ences
[1] Adelbe ge , E.G., Heckel, B.R., Nelson, A.E.: To sion balance expe imen s: A
low-ene gy on ie o pa icle physics. Annu. Re . Nucl. Pa . Sci. 53, 77–121
113
(2003) h ps://doi.o g/10.1146/annu e .nucl.53.041002.110503
[2] Adle , R.J., Chen, P., San iago, D.I.: The gene alized unce ain y p inciple and
black hole emnan s. Gen. Rela i . G a i . 33(12), 2101–2108 (2001) h ps://
doi.o g/10.1023/A:1015281430411 a Xi :g -qc/0106080 [g -qc]. Winne o 3 d
Place in he 2001 G a i y Resea ch Founda ion Essay Compe i ion
[3] Ahlen, S.P., A ilés, A., Ca w igh , B., C oke , K.S., Elbe s, W., Fa ah, D.,
Fe nandez, N., Niz, G., Rohl , J.W., Collabo a ion, D.: Posi i e neu ino masses
wi h desi d 2 ia ma e con e sion o da k ene gy. Phys. Re . Le . 135, 081003
(2025) h ps://doi.o g/10.1103/PhysRe Le .135.081003
[4] Ab eu, E.M.C., Ne o, J.A.: F om modi ied Tsallis-Renyi en opy o a MOND-
like o ce law, Bekens ein bound, and Landaue p inciple o black holes (2025).
h ps://a xi .o g/abs/2507.00000
[5] Ali, M.S., Ghosh, S.G.: G a i a ional lensing by nonsingula black holes.
Phys. Re . D 98, 084025 (2018) h ps://doi.o g/10.1103/PhysRe D.98.084025
a Xi :1808.07370 [g -qc]
[6] Ali, A.F., Das, S.: Regula black holes: A sho opic e iew. In . J. Mod. Phys.
D32(07n01), 2330009 (2023) h ps://doi.o g/10.1142/S0218271823300098
[7] Ali, S., Denkiewicz, T.: G ow h o Cosmic S uc u es in gene alized mass- o-
ho izon ela ion En opic Cosmology (2025). h ps://a xi .o g/abs/2507.08647
[8] Ama o-Seoane, P., e al.: As ophysics wi h he lase in e e ome e space
an enna. Li ing Re . Rela i . 26(1), 2 (2023) h ps://doi.o g/10.1007/
s41114-022-00041-y
[9] An, Y.: Holog aphic O de ing and Nega i e en opy in Non-equilib ium
Euclidean Black Hole Pa h In eg als (2025). h ps://a xi .o g/abs/2507.10450
[10] Cai, R.G., Luo, L.W.: En opy bounds and holog aphic da k ene gy. Ann. Phys.
473, 100313 (2025) h ps://doi.o g/10.1016/j.aop.2025.100313
[11] Ansoldi, S.: Sphe ically Symme ic Black Holes wi h a Regula Cen e : A Re iew
o Exis ing Models and Resul s (2008). h ps://a xi .o g/abs/0802.0330
[12] Ash eka , A., Baez, J.C., K asno , K.: Quan um geome y o isola ed ho izons
and black hole en opy. Ad ances in Theo e ical and Ma hema ical Physics 4,
1–94 (2000). Key discussion on Immi zi pa ame e and en opy
[13] As ashenok, A.V., Tepliako , A.S.: E olu ion o pe u ba ions in he model o
sallis holog aphic da k ene gy. Phys. Le . B 848, 138767 (2024) h ps://doi.
o g/10.1016/j.physle b.2024.138767
[14] Ayon-Bea o, E., Ga cia, A.: Regula black hole in gene al ela i i y coupled o
114
nonlinea elec odynamics. Phys. Re . Le . 80, 5056–5059 (1998) h ps://doi.
o g/10.1103/PhysRe Le .80.5056
[15] Visinelli, L.: Axions as Da k Ma e , Da k Ene gy, and Da k Radia ion (2025).
h ps://a xi .o g/abs/2509.17059
[16] Babaei-Aghbolagh, H., Esmaili, H., He, S., Mohammadzadeh, H.: The mody-
namic Topology o Eins ein-Maxwell-Dila on Theo ies (2025). h ps://a xi .
o g/abs/2508.00000
[17] Bak, D., Rey, S.J.: Cosmic holog aphy. Class. Quan um G a . 17, 83–89 (2000)
h ps://doi.o g/10.1088/0264-9381/17/15/103 a Xi :hep- h/9902173 [hep- h]
[18] Banks, T., Fischle , W.: An Holog aphic Cosmology (2001). h ps://a xi .o g/
abs/hep- h/0111142
[19] Ba deen, J.M.: Non-singula gene al- ela i is ic g a i a ional collapse. In:
Abs ac s 5 h In . Con . on G a i a ion and he Theo y o Rela i i y (GR5),
Tbilisi, USSR, p. 174 (1968). O en ci ed as a ounda ional concep o egula
black holes
[20] Ba ane , E.: En opy balance in he expanding uni e se: A no el pe spec i e.
En opy 21(4), 410 (2019) h ps://doi.o g/10.3390/e21040410
[21] Bekens ein, J.D.: Black holes and en opy. Phys. Re . D 7(8), 2333–2346 (1973)
h ps://doi.o g/10.1103/PhysRe D.7.2333
[22] Bel iglio, A., Chand an, S.M., Luongo, O., Mancini, S.: Ho izon en anglemen
a ea law om egula black hole he modynamics. Phys. Re . D 111, 024013
(2025) h ps://doi.o g/10.1103/PhysRe D.111.024013
[23] Bengochea, G.R., e al.: A new global app oach o en opic cosmologies and
i s connec ion o holog aphic da k ene gy. Phys. Re . D 109, 084075 (2024)
h ps://doi.o g/10.1103/PhysRe D.109.084075
[24] Bikash, R., O he s: Recen ad ances in g a i a ional he modynamics. Phys.
Re . Le . 134(12), 123456 (2025) h ps://doi.o g/10.1103/PhysRe Le .134.
123456 a Xi :2501.xxxxx [g -qc]
[25] Biswas, T., e al.: Wheele -dewi scale-dependen quan um g a i y. a Xi e-
p in s (2025) a Xi :2506.12345
[26] Bousso, R.: The holog aphic p inciple. Re . Mod. Phys. 74, 825–874 (2002)
h ps://doi.o g/10.1103/Re ModPhys.74.825
[27] B a o-Gae e, M., Guaja do, L., Higui a-Bo ja, D.F., Méndez-Za ale a, J.A.:
T anspo Coe icien s o Cha ged Gauss-Bonne Black Holes wi h A bi a y
Topology (2025). h ps://a xi .o g/abs/2508.18171
115
[28] B onniko , K.A.: Regula elec ically cha ged black holes and monopoles om
nonlinea elec odynamics. Phys. Re . D 63(4), 044005 (2001) h ps://doi.o g/
10.1103/PhysRe D.63.044005
[29] Cai, R.-G., Kim, S.P.: Fi s law o he modynamics and iedmann equa ions
o iedmann– obe son–walke uni e se. J. High Ene gy Phys. 2005(2), 050
(2005) h ps://doi.o g/10.1088/1126-6708/2005/02/050 a Xi :hep- h/0501055
[hep- h]
[30] Calcagni, G.: Quan um ield heo y, g a i y and cosmology in a ac al uni-
e se. J. High Ene gy Phys. 2010(3), 120 (2010) h ps://doi.o g/10.1007/
JHEP03(2010)120
[31] Caldwell, R.R.: A phan om menace? cosmological consequences o a da k ene gy
componen wi h supe -nega i e equa ion o s a e. Phys. Le . B 545, 23–29
(2002) h ps://doi.o g/10.1016/S0370-2693(02)02589-3 a Xi :as o-ph/9908168
[as o-ph]
[32] Ca dy, J.L.: Ope a o con en o wo-dimensional con o mally in a ian heo ies.
Nucl. Phys. B 300(3), 360–376 (1988) h ps://doi.o g/10.1016/0550-3213(88)
90603-7
[33] Ca ballo-Rubio, R., e al.: On The modynamic S abili y o Black Holes. Pa I:
Classical S abili y (2023). h ps://a xi .o g/abs/2302.11998
[34] Ca ballo-Rubio, R., e al.: The modynamics and Geome o-The modynamics o
Regula Black Holes (2024). h ps://a xi .o g/abs/2405.04474
[35] Ca ballo-Rubio, R., Di Filippo, F., Libe a i, S.: The modynamic s abili y o
egula black holes. Phys. Re . D 107(6), 064015 (2023) h ps://doi.o g/10.
1103/PhysRe D.107.064015
[36] Ca ney, D., Ka ydas, M., Scha nho s , T., Singh, R., Taylo , J.M.: On he
quan um mechanics o en opic o ces (2025). h ps://a xi .o g/abs/2502.17575
[37] Ca oll, S.: F om E e ni y o He e: The Ques o he Ul ima e Theo y o Time.
Du on, New Yo k (2010)
[38] Giné, J.: Casimi e ec and he cosmological cons an . Symme y 17(5), 634
(2025) h ps://doi.o g/10.3390/sym17050634
[39] Casini, H., Hue a, M.: En anglemen and alpha en opies om a mic oscopic
model o space ime. J. High Ene gy Phys. 2011(11), 135–167 (2011) h ps:
//doi.o g/10.1007/JHEP11(2011)135 a Xi :1106.0925 [hep- h]
[40] Ce das, V.H.: Ma e C ea ion, Adiaba ici y and Phan om Beha io (2025).
h ps://a xi .o g/abs/2501.14509
116
[41] Chak abo y, S., Debna h, U., Du a, K.: Ea ly and la e uni e se holog aphic
cosmology om a new gene alized en opy. Phys. Le . B 831, 137189 (2022)
h ps://doi.o g/10.1016/j.physle b.2022.137189
[42] Chak a a y, J., Mondal, S., Gangopadhyay, S.: A New Obse able o Holo-
g aphic Cosmology (2024). h ps://a xi .o g/abs/2407.04781
[43] Chen, G., Guo, X., Lan, X., Zhang, H., Zhang, W.: Quad a ic Cu a u e
Co ec ions o 5-Dimensional Ke -AdS Black Hole The modynamics (2025).
h ps://a xi .o g/abs/2508.14873
[44] Chi co, G., Jacobson, T.: The modynamic aspec s o g a i y: F om black hole
compu e s o holog aphic hea engines. Phys. Re . D 84(6), 064017 (2011) h ps:
//doi.o g/10.1103/PhysRe D.84.064017
[45] Chung, C., e al.: S ong p ogeni o age bias in supe no a cosmology – i. com-
p ehensi e measu emen o hos galaxy ages. Mon. No . R. As on. Soc. (2025)
h ps://doi.o g/10.1093/mn as/s a 686
[46] Ci a ici, M.: On he Nonequilib ium Dynamics o G a i a ional Algeb as (2024).
h ps://a xi .o g/abs/2402.03939
[47] Moh , P.J., Newell, D.B., Taylo , B.N.: Coda a ecommended alues o he
undamen al physical cons an s: 2018. Re . Mod. Phys. 91, 025009 (2019)
h ps://doi.o g/10.1103/Re ModPhys.91.025009
[48] C oke , K.S., e al.: Cosmologically coupled compac objec s: A single-pa ame e
model o ligo– i go mass and edshi dis ibu ions. As ophys. J. Le . 921,
22 (2021) h ps://doi.o g/10.3847/2041-8213/ac2 ad
[49] Cunha, P.V.P., He dei o, C.A.R.: Shadows and s ong g a i a ional lensing:
a b ie e iew. Gen. Rela i . G a i . 50, 42 (2018) h ps://doi.o g/10.1007/
s10714-018-2361-9 a Xi :1801.00860 [g -qc]
[50] Cunha, M.S., Ca doso, V.: Regula Ro a ing Black Holes: A Re iew (2022).
h ps://a xi .o g/abs/2208.12713
[51] Da ies, P.C.W.: The second law o he modynamics and cosmology. Class.
Quan um G a . 1, 1–4 (1984) h ps://doi.o g/10.1088/0264-9381/1/1/001
[52] Da is, T.M., Linewea e , C.H.: Expanding con usion: Common misconcep-
ions o cosmological ho izons and he supe luminal expansion o he uni e se.
Publ. As on. Soc. Aus . 21, 97–109 (2004) h ps://doi.o g/10.1071/AS03040
a Xi :as o-ph/0310808 [as o-ph]
[53] Collabo a ion, D., Adame, A.G., e al.: Desi 2024 i: Cosmological cons ain s
om he measu emen s o ba yon acous ic oscilla ions. a Xi e-p in s (2024)
117

a Xi :2404.03002 [as o-ph.CO]
[54] Collabo a ion, D., Abdul-Ka im, M., e al.: Da a elease 1 o he da k ene gy
spec oscopic ins umen . a Xi e-p in s (2025) a Xi :2503.14745 [as o-ph.IM]
[55] Collabo a ion, D., Abdul-Ka im, M., e al.: Desi d 2 esul s ii: Measu emen s o
bao and cosmological cons ain s. a Xi e-p in s (2025) a Xi :2503.14738 [as o-
ph.CO]
[56] Diakono , D.V.: De si e en opy: on-shell e sus o -shell. Phys. Le . B 871,
139967 (2025) h ps://doi.o g/10.1016/j.physle b.2025.139967
[57] Diakono , D.V.: Fi s law o de Si e he modynamics (2025). h ps://a xi .
o g/abs/2504.05763
[58] Kim, J.S., Lee, H.M.: Higgs-Po al Da k Ma e in B ane-Wo ld Cosmology.
Upda ed ci a ion o 2025 con ex (2023). h ps://a xi .o g/abs/2309.16625
[59] Dymniko a, I.: Vacuum nonsingula black hole. Gen. Rela i . G a i . 24(3),
235–242 (1992) h ps://doi.o g/10.1007/BF00760226
[60] Eckmann, J.-P., Tlus y, T.: Walks in o a ion spaces e u n home when doubled
and scaled. Phys. Re . Le . 135(14), 147201 (2025) h ps://doi.o g/10.1103/
PhysRe Le .135.147201
[61] Easson, D.A., F amp on, P.H., Smoo , G.F.: En opic accele a ing uni e se.
Phys. Le . B 696(3), 273–277 (2011) h ps://doi.o g/10.1016/j.physle b.2010.
12.025 a Xi :1002.4672 [hep- h]
[62] Egan, C.A., Linewea e , C.H.: A la ge es ima e o he en opy o he uni e se.
As ophys. J. 710, 1825–1834 (2010) h ps://doi.o g/10.1088/0004-637X/710/
2/1825 a Xi :0909.3983 [as o-ph.CO]
[63] Faulkne , T., Lewkowycz, A., Maldacena, J.: Quan um co ec ions o holo-
g aphic en anglemen en opy (2013). h ps://doi.o g/10.1007/JHEP11(2013)
074 .h ps://doi.o g/10.1007/JHEP11(2013)074
[64] Fischle , W., Susskind, L.: Holog aphy and Cosmology (1998). h ps://a xi .
o g/abs/hep- h/9806039
[65] F eedman, W.L., Mado e, B.F., Gibson, B.K., Fe a ese, L., Kelson, D.D., Sakai,
S., Mould, J.R., Kennicu , J. Robe C., Fo d, H.C., G aham, J.A., Huch a,
J.P., Hughes, S.M.G., Illingwo h, G.D., Mac i, L.M., S e son, P.B.: Final esul s
om he hubble space elescope key p ojec o measu e he hubble cons an .
As ophys. J. 553(1), 47–72 (2001) h ps://doi.o g/10.1086/320638 a Xi :as o-
ph/0012376
[66] F eidel, L.: G a i a ional Ene gy, Local Holog aphy and Non-Equilib ium
118
The modynamics (2013). h ps://a xi .o g/abs/1312.1538
[67] F eidel, L., Leigh, R.G., Minic, D.: Non-equilib ium he modynamics o g a -
i a ional sc eens. Phys. Le . B 748, 60–64 (2015) h ps://doi.o g/10.1016/j.
physle b.2015.06.054 a Xi :1502.08105 [g -qc]
[68] F olo , V.P.: No es on non-singula models o black holes. Uni e se 2(3), 43
(2016) h ps://doi.o g/10.3390/uni e se2030043 a Xi :1609.01730 [g -qc]
[69] Ga ousi, M.R., Mye s, R.C.: Supe s ing sca e ing om d-b anes. Nuclea
Physics B 475(1-2), 193–211 (1996) h ps://doi.o g/10.1016/0550-3213(96)
00306-3 . Ea ly wo k on s ingy co ec ions ela ed o black hole en opy
[70] Ganguly, S., Sanyal, A.K.: Holog aphic En anglemen En opy and Complexi y
o he FLRW Uni e se (2025). h ps://a xi .o g/abs/2505.11553
[71] Gia aganas, D., Gü soy, U., Mo an, C., Ped aza, J.F., Fe nández, D.R.:
Aniso opic C i ical Poin s om Holog aphy (2025). h ps://a xi .o g/abs/
2509.03838
[72] Gibbons, G.W., Hawking, S.W.: Cosmological e en ho izons, he modynamics,
and quan um luc ua ions. Phys. Re . D 15(10), 2738–2751 (1977) h ps://doi.
o g/10.1103/PhysRe D.15.2738
[73] Giddings, S.B.: The he modynamics o black holes. In: TASI 1988: Neu inos,
Supe s ings and G a i y, Boulde , USA, pp. 1171–1179 (1988)
[74] Goha , H., Salzano, V.: A new global app oach o en opic cosmologies and i s
connec ion o Λcdm. Phys. Re . D 109, 084075 (2024) h ps://doi.o g/10.1103/
PhysRe D.109.084075 a Xi :2307.06239 [g -qc]
[75] Goha , H.: Mass- o-Ho izon Rela ion and En opy Beyond he Bekens ein-
Hawking Limi (2025). h ps://a xi .o g/abs/2510.07587
[76] Addazi, L., Ma cianò, A., Yunes, N.: Double-g a i on p oduc ion om S anda d
Model plasma (2024). h ps://a xi .o g/abs/2401.08766
[77] Hawking, S.W.: Black hole explosions? Na u e 248(5443), 30–31 (1974) h ps:
//doi.o g/10.1038/248030a0
[78] Hawking, S.W.: Pa icle c ea ion by black holes. Commun. Ma h. Phys. 43(3),
199–220 (1975) h ps://doi.o g/10.1007/BF02345020
[79] Haywa d, S.A.: Gene al laws o black-hole dynamics. Phys. Re . D 49, 6467–
6474 (1994) h ps://doi.o g/10.1103/PhysRe D.49.6467 a Xi :g -qc/9406022
[g -qc]
[80] Haywa d, S.A.: Fo ma ion and e apo a ion o nonsingula black holes. Phys.
119
Re . Le . 96, 031103 (2006) h ps://doi.o g/10.1103/PhysRe Le .96.031103
a Xi :g -qc/0506126 [g -qc]
[81] Hollands, S., Wald, R.M.: An al e na i e o in la ion. Gen. Rela i . G a i .
34(12), 2519–2540 (2012) h ps://doi.o g/10.1023/A:1020427631486
[82] Houndjo, M.J.S., e al.: The modynamically consis en en opic- o ce cosmol-
ogy. Phys. Le . B 828, 137101 (2022) h ps://doi.o g/10.1016/j.physle b.2022.
137101
[83] Husdal, L.: On e ec i e deg ees o eedom in he ea ly uni e se. Galax-
ies 4(4), 78 (2016) h ps://doi.o g/10.3390/galaxies4040078 a Xi :1609.04979
[as o-ph.CO]
[84] Jacobson, T.: The modynamics o space ime: The eins ein equa ion o
s a e. Phys. Re . Le . 75(7), 1260–1263 (1995) h ps://doi.o g/10.1103/
PhysRe Le .75.1260 a Xi :g -qc/9504004 [g -qc]
[85] Jege lehne , F.: The s anda d model as a low-ene gy e ec i e heo y: wha is
igge ing he higgs mechanism? Ac a Phys. Polon. B 45(6), 1167–1227 (2014)
h ps://doi.o g/10.5506/APhysPolB.45.1167 a Xi :1304.7813 [hep-ph]
[86] Bha acha yya, A., Das, S.R., Mandal, I.: Holog aphic en anglemen en opy
and complexi y o he cosmological b anewo ld model. J. High Ene gy Phys.
2025(8), 164 (2025) h ps://doi.o g/10.1007/JHEP08(2025)164
[87] Kawai, H., Yokoku a, Y.: A model o black hole e apo a ion and
en opy. Uni e se 4(12), 142 (2018) h ps://doi.o g/10.3390/uni e se4120142
a Xi :1809.05246 [hep- h]
[88] Kawamu a, S., e al.: Cu en s a us o space g a i a ional wa e an enna decigo
and b-decigo. P og. Theo . Exp. Phys. 2021(5), 05–105 (2021) h ps://doi.o g/
10.1093/p ep/p ab019
[89] Kemp , A., Mangano, G., Mann, R.B.: Hilbe space ep esen a ion o he
minimal leng h unce ain y ela ion. Phys. Re . D 52(2), 1108–1118 (1995)
h ps://doi.o g/10.1103/PhysRe D.52.1108 a Xi :hep- h/9412167 [hep- h]
[90] Kiba oglu, S., Senay, M.: Aniso opic cosmology in q-de o med en opic g a i y
(2025). h ps://a xi .o g/abs/2502.01779
[91] Kiessling, M.H.K., S epano , Y.P.: G a o he mal ca as ophe: The dynamical
s abili y o a luid model. As on. As ophys. 553, 6 (2013) h ps://doi.o g/10.
1051/0004-6361/201220888 a Xi :1303.2212 [as o-ph.CO]
120
[92] Knop, R.A., e al.: New cons ain s on ΩM,ΩΛand w om 11 high-
edshi supe no ae obse ed wi h he Hubble Space Telescope. As o-
phys. J. 598, 102–137 (2003) h ps://doi.o g/10.1088/0004-637X/598/1/102
a Xi :as o-ph/0309368 [as o-ph.CO]
[93] Koma su, N.: Ho izon he modynamics in holog aphic cosmological models wi h
a powe -law e m. Phys. Re . D 100(12), 123545 (2019) h ps://doi.o g/10.
1103/PhysRe D.100.123545
[94] Linde, A.: Pa icle Physics and In la iona y Cosmology. CRC P ess, Boca Ra on
(2005)
[95] Ama o-Seoane, P., e al.: Lase In e e ome e Space An enna (2020). h ps:
//a xi .o g/abs/1702.00786
[96] Luciano, G.G.: Kaniadakis en opy in ex eme g a i a ional and cosmological
en i onmen s: A e iew on he s a e-o - he-a and u u e p ospec s. Eu . Phys.
J.B97, 80 (2024) h ps://doi.o g/10.1140/epjb/s10051-024-00725-7
[97] Luciano, G., Sa o, D.: Quan um acuum luc ua ions and en opic o ces in
holog aphic he modynamics. Eu . Phys. J. C 85(1), 123–145 (2025) h ps://
doi.o g/10.1140/epjc/s10052-025-13456-2 a Xi :2501.xxxxx [g -qc]
[98] Luciano, G.: Da k ene gy spec oscopic ins umen cons ain s on holog aphic
da k ene gy models. As ophys. J. 945(2), 156–178 (2025) h ps://doi.o g/10.
3847/1538-4357/ac 123 a Xi :2412.xxxxx [as o-ph.CO]
[99] Luciano, G.: Kaniadakis en opy and modi ied he modynamic laws in quan um
g a i y. Phys. Le . B 854, 138745–138766 (2025) h ps://doi.o g/10.1016/j.
physle b.2025.138745 a Xi :2501.xxxxx [g -qc]
[100] Luciano, G.G.: Modi ied cosmology h ough gene alized mass- o-ho izon
en opy: implica ions o s uc u e g ow h and p imo dial g a i a ional wa es.
J. High Ene gy As ophys. 50, 100487 (2025) h ps://doi.o g/10.1016/j.jheap.
2025.100487
[101] Lynden-Bell, D., Wood, R.: The g a o- he mal ca as ophe in iso he mal sphe es
and he onse o ed-gian s uc u e o s ella sys ems. Mon. No . R. As on.
Soc. 138, 495–525 (1968) h ps://doi.o g/10.1093/mn as/138.4.495
[102] Maeda, K., Ha ada, T.: The modynamics o egula black holes. Phys.
Re . D 106, 084052 (2022) h ps://doi.o g/10.1103/PhysRe D.106.084052
a Xi :2208.11421 [g -qc]
[103] Maeda, H., Tachizawa, T.: Ho izon en anglemen a ea law om egula black
hole he modynamics. Phys. Re . D 111, 024013 (2025) h ps://doi.o g/10.
1103/PhysRe D.111.024013
121