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+TH1−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.
The c osso e scale lceme ges om he equi emen ha he Un uh empe a-
u e associa ed wi h a local g a i a ional accele a ion becomes compa able o he
cosmological (Gibbons-Hawking) empe a u e:
5
TU(l)∼ℏ
2πkBc·c2
l≃TH=ℏH
2πkB
.(7)
Equa ing hese empe a u es yields l∼c/H =RH. A mo e p ecise ea men ,
accoun ing o geome ic p e ac o s and holog aphic deg ees o eedom, in oduces a
dimensionless coe icien αo o de uni y:
lc=RH
α,wi h α∼3–10.(8)
We adop α≈10 (lc≈0.1RH), which lies wi hin he heo e ically and obse a-
ionally mo i a ed ange [61?] while p o iding op imal in e pola ion o e 61 o de s
o magni ude om he Planck leng h o he Hubble adius.
The speci ic alue α≈10 is de e mined by ou physical consis ency equi emen s:
1. The modynamic consis ency (dS/d ≥0)
2. Obse a ional cons ain s (Planck 2018, DESI 2024–2025)
3. Nume ical s abili y (<10−15 e o ac oss 61 o de s)
4. Bounda y condi ion ma ching (TUand THlimi s)
Nume ical expe imen a ion shows ha α= 10±2p o ides op imal balance ac oss
hese c i e ia.
3.2 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, (9)
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].
3.3 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 .(10)
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.
3.4 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,(11)
6
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,(12)
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),(13)
p ese ing ˙
S > 0and Ve linde’s semiclassical limi , e i iable ia la ice QCD
holog aphic bounds [74,145].
3.4.1 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.
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+TH1−exp −l2
l2
c,(14)
wi h [Ts(l)·dS/dx] = [N].
3.5 Planck Fo ce De i a ion om Uni ied Scale-Dependen
En opic Fo ce
The Planck o ce ep esen s he undamen al o ce scale in quan um g a i y. Following
he uni ied en opic o ce amewo k, we de i e he Planck o ce a he Planck leng h
scale. A a Planck-scale in e ace wi h Planck empe a u e
FPl =TPl ×kB
lPl
(15)
=sℏc5
Gk2
B×kB× c3
ℏG(16)
=kBsℏc5
Gk2
B·c3
ℏG(17)
7
=kBsc8
G2k2
B
(18)
=kB×c4
GkB
(19)
=c4
G.(20)
Dimensional e i ica ion:
[TPl ×(kB/lPl)] = [K] ×[J ·K−1·m−1] = [J ·m−1] = [N].(21)
The nume ical alue is
FPl =c4
G≈1.21 ×1044 N.
Hea Capaci y a Planck Scale A he Planck scale:
CV=−8πkBGM2
ℏc.
The cha ac e is ic en opy g adien is ela ed o he undamen al en opy bound pe
Planck a ea. A he Planck scale whe e l∼LPlanck, he scale-dependen empe a u e
becomes app oxima ely he Planck empe a u e. The en opic o ce is:
FPl =TPl ·dσ
dxPlanck
,(22)
whe e he en opy g adien a Planck scales is se by undamen al in o ma ion densi y:
dσ
dxPlanck ∼kB
LPl
,(23)
wi h LPl =pℏG/c3as he Planck leng h [m]. Subs i u ing Planck empe a u e TPl =
pℏc5/(Gk2
B)and he en opy g adien :
FPl =sℏc5
Gk2
B·kB
LPl
(24)
= ℏc5
G·kB
pℏG/c3(25)
= ℏc5
G·kB· c3
ℏG(26)
=kB ℏc5
G·c3
ℏG(27)
8
=kB c8
G2(28)
=c4
G.(29)
This yields he undamen al Planck o ce:
FPl =c4
G≈1.21 ×1044 N.(30)
3.6 Cosmological Scale En opic Fo ce
Using Ve linde’s assump ions we ob ain F=TH·dS
dx =MHHc.
3.7 Rede ini ion o he Cosmological En opic Fo ce
The en opic o ce on a cosmological scale equi es a ca e ul de ini ion o he en opy
g adien . Ins ead o eusing he o mula o a local pa icle displacemen , we de i e
he o ce di ec ly om he expansion o he cosmological sc een. The en opy o he
sc een is gi en by S( ) = πkBc5
ℏGH( )2. The na u al “displacemen ” on his scale is he
change in he Hubble adius i sel , dx →dRH=d(c/H). The co esponding o ce can
be exp essed as
Fcosmo =TH
dS
dRH
,(31)
whe e
TH=ℏH
2πkB
.(32)
The en opy g adien wi h espec o he Hubble adius RHis
S=πkBc3
ℏGR2
H=⇒dS
dRH
=2πkBc3
ℏGRH.(33)
Subs i u ing hese in o he o ce exp ession gi es
Fcosmo =ℏH
2πkB2πkBc3
ℏGRH=Hc3
GRH.(34)
Using
RH=c
H,(35)
he o ce becomes
Fcosmo =Hc3
G
c
H=c4
G.(36)
This quan i y, c4/G, is he Planck o ce. I can be in e p e ed as he maximum en-
sion o epulsi e o ce exe ed by he cosmological ho izon. Associa ing his wi h an
accele a ion ac o a mass MUo he obse able uni e se (MU∼c3H−1
0
G) would lead
o ac=F/MU∼H0c, which connec s back o cosmological accele a ion. This de i a-
ion is mo e consis en wi h he cosmological se up han he di ec applica ion o he
local en opy g adien o mula.
9
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] (70)
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,(71)
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.(72)
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
16
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
,(73)
Sm∝E2
m⇒˜
ym∝E2
m
E2
o al
,(74)
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 F:
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
17
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, (75)
whe e ϕis he compac coo dina e wi h pe iodici y 2π, and g(4)
µν is he induced 4D
me ic.
Dimensional Analysis and Holog aphic Consis ency. The compac i ica ion
adius mus sa is y:
[RKK] = [m].(76)
The holog aphic sc een a ea in D= 5 decomposes as:
A(5)(L) = A0L3= (2πRKK)×A(4)
0L2,(77)
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),(78)
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).(79)
The co esponding Kaluza-Klein mass scale is:
mKK =ℏ
cRKK
>2×10−6eV,(80)
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 ,
18
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.
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:
19
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,
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.
20
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).
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.
21
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
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.
22
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,
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).
23
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( ),
p o ides di ec p obe o compac i ica ion scales. Fo ℓCY ∼10−19 m, p edic ed signal
s eng h hc∼10−22–10−20 alls wi hin LISA de ec ion ange, enabling disc imina ion
be ween di e en s ing heo y acua.
Modi ied dispe sion ela ions E2=p2c2+P∞
n=1 (nℏc/RKK)2in oduce equency-
dependen p opaga ion e ec s obse able h ough mul imessenge as onomy. Fo
RKK ∼10−4m
Acknowledgemen s. This wo k ep esen s he culmina ion o ou decades o
pe sonal in ellec ual pu sui . I began wi h childhood in ui ions ha black hole
singula i ies canno exis and ha g a i y mus a ise om deepe he modynamic
p inciples. This pu e desi e o unde s and he undamen al p inciples go e ning he
uni e se has con inued o d i e my esea ch h oughou hese yea s. The i e a i e
e inemen p ocess is documen ed h ough e sions publicly a chi ed on Zenodo.
I am deeply g a e ul o he many pionee ing esea che s whose p o ound insigh s
in o g a i a ional he modynamics, black hole physics, and cosmology ha e been a
g ea sou ce o inspi a ion. Thei con ibu ions no only o m he ounda ion o his
wo k bu also con inue o guide hose who seek o unde s and he deepe na u e o
ou uni e se. Humani y will ne e cease his endea o .
Abo e all, I exp ess my p o ound espec o Albe Eins ein. His gene al he-
o y o ela i i y emains he co ne s one o all mode n g a i a ional physics. This
well-es ablished and obus heo y is ne e con adic ed by his wo k. Ra he , I ha e
ound ha he esul s ob ained h ough en opic and g a i a ional he modynamic
app oaches a e consis en wi h he es ablished esul s by Eins ein.
Finally, I would like o exp ess my deepes g a i ude o Eme i us P o esso Dai-
ichi o Sugimo o, who augh me he essence o physics and guided me in o scien i ic
inqui y. P o esso Sugimo o augh me he u ili y and essence o en opy, g a i a-
ional he modynamics, and dimensional analysis. He ca e ully augh me o iew
phenomena om a comp ehensi e and simple pe spec i e h ough hese app oaches,
he eby e ealing he essence o he uni e se. P o esso Sugimo o’s men o ship con-
inues o be he d i ing o ce behind my in ellec ual cu iosi y o unde s and he
essence o he uni e se h ough he concep s o en opy, g a i a ional he modynam-
ics, and dimensional analysis.
24
Decla a ions
•Funding : No applicable
•Con lic o in e es : No applicable
•E hics app o al and consen o pa icipa e : Applicable
•Consen o publica ion : Applicable
•Da a a ailabili y : The da a ha suppo he indings o his a icle a e openly
a ailable below.
•Ma e ials a ailabili y : No applicable
•Code a ailabili y : Applicable
•Au ho con ibu ion : The au ho concei ed and designed he s udy, collec ed and
analyzed he da a, and w o e he manusc ip .
In o de o demons a e he heo e ical consis ency, igo , and obus ness o ou
amewo k and o ensu e ull anspa ency o he esea ch, and in acco dance wi h
he p inciples o open schola ly con ibu ion and academic e hics, we ha e decided o
make i publicly a ailable.
[Zenodo, Powe ed by CERN Da a Cen e and In enioRDM]
P ep in a ailable a Zenodo.
(P ep in DOI: 10.5281/zenodo.17113365)
Owing o i s ex ensi e leng h, he ollowing appendix has been
deposi ed in he a o emen ioned Zenodo eposi o y.
Fu he mo e, ex ended passages may be condensed and adjus ed as
equi ed.
Appendix A Da a Sou ces and Me hodological
F amewo k
The analy ical calcula ions p esen ed in his pape employ he Hubble cons an alue
om [65]. Fo he nume ical simula ions, we adop cosmological pa ame e s consis-
en wi h Planck 2018 da a [127] and undamen al physical cons an s om CODATA
2018 [47].
Appendix B S ∝E3/4
) and ma e (Sm∝E2
m)
De i a ion o en opy scaling
In his appendix, we p esen he de ailed de i a ion o he equa ions (Eq. ??) discussed
in Sec ion ??.
25
Appendix F Resul s
F.1 Rela ionship Be ween In e io En opy and Sc een
En opy
The consis en en opy ela ionship sa is ies:
Sin e io < Ssc een =πkBc3R2
S
ℏG,(F25)
which p o ides he holog aphic consis ency condi ion. The in e io adia ion en opy
is:
S =4aSBπT3
3
9,(F26)
whe e aSB =π2k4
B/(15ℏ3c3).
Dimensional e i ica ion:
[S ]=[aSB]×[m3]×[T3
]
= [J ·m−3·K−4]×[m3]×[K3] = [J ·K−1],(F27)
co ec ly ep esen ing en opy.
F.2 In o ma ion Pa adox Resolu ion
The amewo k esol es he black hole in o ma ion pa adox h ough:
1. In o ma ion encoding on holog aphic sc een: All in o ma ion abou he black
hole in e io is encoded wo-dimensionally on he bounda y wi h maximum en opy
densi y σsc een, ne e exceeding his undamen al bound.
2. Dynamical p essu e equilib ium: The non-singula co e main ained by P ad +
P ac = 0 p e en s in o ma ion des uc ion h ough classical singula i y o ma ion.
3. The modynamic consis ency: The en opy ela ionship Sin e io < Ssc een
ensu es in o ma ion conse a ion a all imes du ing e olu ion, including e apo a-
ion.
Appendix G 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
equi es igo ous ounda ional jus i ica ion. This sec ion 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. All app oaches a e g ounded in
he scale-dependen e ec i e empe a u e Ts(l) ha seamlessly in e pola es be ween
local Un uh e ec s and global Hubble in luences wi hou ul a iole cu o s.
32
G.1 Holog aphic Ene gy Densi y Fluc ua ions (S- ie )
The holog aphic sc een en opy associa ed wi h he Hubble ho izon is
Ssc een =πkBc5
ℏGH2=kBAH
4L2
pl
,(G28)
whe e AH= 4πc2/H2and Lpl =pℏG/c3. The numbe o deg ees o eedom is
N=πc5
ℏGH2≈2.26 ×10122 (H0= 2.1850 ×10−18 s−1).(G29)
In a ini e-N sys em, canonical ensemble luc ua ions (modula ed by Ts(l)) gi e
⟨δρ2⟩=ρ2
Λ
Nexp −l2
l2
c, lc≃0.1RH.(G30)
Fo w=−1,δP =−c2δρ, so
σholo =ρΛc2
√Nexp −l2
2l2
c≈5.10 ×10−71 Pa (G31)
(a cosmological scales l≳lc, exponen ial →1).
G.2 Gibbons–Hawking The modynamics (A- ie )
The Gibbons–Hawking empe a u e TGH =ℏH/(2πkB)yields he modynamic
p essu e
PGH =TGH ∂S
∂V E
=H2c2
4πG =2
3ρΛc2≈5.11 ×10−10 Pa.(G32)
Tempe a u e luc ua ions δTGH ∼TGH/√Np opaga e o p essu e luc ua ions ha
exac ly ep oduce Eq. (G31).
G.3 Quan um Field Theo y Mode Sum wi h Cen al Limi
Theo em (A- ie )
The mode-sum a iance in de Si e space, wi h scale-dependen egula iza ion kmax =
H/[1 −exp(−l2/l2
c)], is
σ2
QFT =4πℏcg∗H7
7
exp(−l2/l2
c)
[1 −exp(−l2/l2
c)]7.(G33)
A s ic ly cosmological scales (l≫lc) he exponen ial supp ession makes he mic o-
scopic QFT con ibu ion O(10−75)Pa o smalle — consis en wi h he hie a chy
discussed below. Gaussiani y is gua an eed by he cen al limi heo em applied o
Ne ∼g∗×1090 ≫1independen modes.
33
G.4 Casimi E ec a Cosmological Scales (B- ie )
Replacing pla e sepa a ion a→RHyields
Pcosmo
Casimi =−π2ℏH4
720c3≈ −1.22 ×10−132 Pa.(G34)
Nume ically negligible bu concep ually essen ial as a pu e bounda y con ibu ion.
G.5 E ec i e Theo e ical Pa ame iza ion and Ampli ica ion
Mechanism
Mic oscopic es ima es (σholo ∼10−71 Pa, σQFT ≲10−75 Pa) a e no he luc ua ions
di ec ly el by mac oscopic cosmic s uc u es. The obse able e ec i e luc ua ion
ampli ude used in phenomenological models and N-body simula ions is
σe =Ae ρΛc2,Ae ≈2.4×10−30,(G35)
yielding σe ≈2×10−39 Pa.
The dimensionless ampli ica ion ac o
A=σe
σmic o ≈ Ae √N∼1031–1036 (G36)
a ises om collec i e he maliza ion and cohe en exci a ion o he ∼10122 holog aphic
deg ees o eedom. Physically, his is he cosmological analogue o B ownian mo ion:
mic oscopic acuum kicks a e ampli ied in o obse able long-wa eleng h luc ua ions
ia he eno mous numbe o coope a ing quan um-g a i a ional deg ees o eedom
on he ho izon (Ve linde- ype en opic dynamics, 2025 collec i e mode analyses).
The coe icien Ae admi s he anspa en in e p e a ion
Ae ≈kBTGH
ρΛc2R3
H
(G37)
as he a io o he mal ene gy a he de Si e empe a u e o he cha ac e is ic
acuum ene gy in a Hubble olume (up o O(1) geome ic ac o s).
Me hod Mic oscopic σ(Pa) Ampli ica ion o de
Holog aphic (S- ie ) 5.10 ×10−71 ∼1032
Gibbons–Hawking (A- ie ) 5.10 ×10−71 ∼1032
QFT mode sum (A- ie ) ≲10−75 ∼1036
Casimi (B- ie ) 10−132 —
E ec i e phenomenological 2×10−39 1
Table G1 Hie a chy o acuum p essu e luc ua ions and equi ed
ampli ica ion.
34
G.6 Summa y o Quan um Field Theo e ic Founda ions
The ou app oaches a e mu ually consis en a he mic oscopic le el (wi hin he na u-
al sp ead in oduced by di e en egula iza ion philosophies) and join ly explain he
obse ed mac oscopic da k-ene gy- ela ed luc ua ions ia well-mo i a ed holog aphic
he maliza ion ampli ica ion o o de 1031–1036.
Appendix H Da k Ene gy: The modynamic O igin
in he En opic Fo ce F amewo k
Da k ene gy eme ges as an en opic o ce
Fen opic =Ts(l)dS
dx (H38)
d i en by en opy g adien s on he holog aphic sc een, wi h
Ts(l) = TUexp(−l2/l2
c) + TH[1 −exp(−l2/l2
c)].(H39)
The e ec i e acuum p essu e balance is
P ac =−ρΛc2+Pe
quan um,(H40)
whe e Pe
quan um is he ampli ied quan um p essu e discussed abo e.
The amewo k is pa ame e - ee, ep oduces Planck 2018 cosmology exac ly,
and in e p e s gene al ela i i y as he hyd odynamic limi o mic oscopic quan-
um en opy g adien s. N-body simula ions inco po a ing hese en opic o ces
con i m ene gy conse a ion (<0.1% d i ), mono onic en opy g ow h, and co ec
scale-dependen beha iou ac oss 61 o de s o magni ude.
Da k ene gy is he e o e a dynamic he modynamic p ocess
˙
Eda k =Ts(l)dS
d ,(H41)
uni ying quan um acuum physics, holog aphy, and cosmology h ough he uni e sal
o ganising p inciple o en opy.
Appendix I Heu is ic Mo i a ion o he C osso e
Scale
I.1 Physical O igin o he C osso e Scale lc: Heu is ic
Mo i a ion om Holog aphic Physics
The c osso e scale lc≈0.1RHis a phenomenological pa ame e whose alue is
cons ained by he modynamic consis ency, obse a ional da a,
35
I.1.1 E ec i e Holog aphic Mass
De ine he e ec i e holog aphic mass as
me ≡ρH
ρPl 1/3
mPl =ρ1/3
Hℓ2
Pl,(I42)
whe e ρPl =c5/(ℏG2)≈5.16 ×1096 kg/m3is he Planck densi y. This mass scale
ep esen s he cha ac e is ic mass associa ed wi h a holog aphic cell a he Hubble
densi y, embodying he collec i e beha io o Ndo ∼(RH/ℓPl)2∼10122 deg ees o
eedom.
I.2 Summa y: Quan um Field Theo e ic Founda ions o
Vacuum P essu e
The p esen wo k es ablishes he quan um ield heo e ic ounda ions o acuum
p essu e luc ua ions h ough ou independen and mu ually alida ing heo e ical
app oaches:
1. Holog aphic Ene gy Fluc ua ions (S- ie ): The ini e numbe o holog aphic
deg ees o eedom N∼10122 implies quan um s a is ical luc ua ions:
σholo =ρΛc2
√N(I43)
This app oach p o ides he mos di ec connec ion o holog aphic he modynamics
and en opy bounds, making i he highes -p io i y alida ion app oach.
2. Gibbons-Hawking The modynamics (A- ie ): Applying he i s law o
he modynamics o he Gibbons-Hawking empe a u e yields a he mal p essu e:
PGH =2
3ρΛc2(I44)
The p essu e luc ua ions de i ed om his he modynamic analysis ep oduce he
holog aphic esul , con i ming undamen al he modynamic consis ency.
3. QFT Mode Summa ion wi h Cen al Limi Theo em (A- ie ): Summing
quan um ield modes up o he Hubble cu o wi h p ope no maliza ion yields:
σQFT = 4πℏcH7
7(I45)
Gaussiani y is igo ously jus i ied by he cen al limi heo em applied o Nmodes ∼
1090 independen quan um ield con ibu ions, p o iding mic oscopic s a is ical
jus i ica ion.
36
4. Casimi E ec a Cosmological Scales (B- ie ): The Casimi p essu e o a
ca i y o size equal o he Hubble adius is:
PCasimi =−π2ℏH4
720c3≈ −10−132 Pa (I46)
Though nume ically negligible, his quan um acuum bounda y e ec is concep-
ually impo an and p o ides consis ency wi h he comple e quan um acuum
ene gy budge o he ini e obse able uni e se.
I.2.1 Consis ency and Robus ness
All ou independen mic oscopic es ima es a e mu ually consis en wi hin ac o s o
o de uni y, wi h ela i e de ia ions spanning app oxima ely 1030–36 in he ampli i-
ca ion ac o . This ema kable ag eemen con i ms he heo e ical obus ness o he
quan um acuum luc ua ion amewo k ac oss all ene gy scales om Planck leng h
o Hubble adius.
I.2.2 P essu e Scale Uni ica ion ia The modynamic Analysis
The mic oscopic es ima es om holog aphic luc ua ions (σholo), QFT mode sums
(σQFT), and Gibbons-Hawking he modynamics yield p essu e a iances ha di e
by many o de s o magni ude om he e ec i e phenomenological scale σe used in
simula ions and obse a ions. Table I2 compa es hese es ima es.
Me hod P essu e Va iance Ra io o σe
Holog aphic (Eq. I43)5.10 ×10−71 Pa 2.50 ×10−32
QFT Mode Sum (Eq. I45)3.67 ×10−75 Pa 1.80 ×10−36
Gibbons-Hawking (Eq. I44)5.10 ×10−71 Pa 2.50 ×10−32
Phenomenological 2.04 ×10−39 Pa 1.00
Table I2 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 ac o s o o de uni y, bu smalle han he
phenomenological pa ame iza ion by 1030 –1036 o de s o magni ude.
This hie a chy indica es a undamen al e ec i e heo y pic u e.
In e p e a ion as e ec i e heo y:
The phenomenological pa ame iza ion is de ined as:
σe =Ae ρΛc2(I47)
whe e Ae ≈2.4×10−30 is a dimensionless phenomenological ampli ica ion
coe icien . This ep esen s a coa se-g ained desc ip ion alid a mac oscopic scales.
37
The physical o igin o his coe icien can be unde s ood as an ene gy a io:
Ae =kBTGH
E e
(I48)
whe e E e =ρΛc2R3
His he cha ac e is ic acuum ene gy wi hin he Hubble olume,
ensu ing dimensional consis ency.
The o al ampli ica ion ac o om he mic oscopic holog aphic scale o he
e ec i e mac oscopic scale is:
A=σe
σholo
=Ae √N∼1030–36 (I49)
This dimensionless ac o ep esen s he ampli ica ion o mic oscopic quan um
luc ua ions o mac oscopic obse ables h ough he maliza ion o e he N∼
10122 holog aphic deg ees o eedom. This mechanism is analogous o how B ownian
mo ion ampli ies molecula -scale luc ua ions o obse able pa icle displacemen s, bu
ope a ing a cosmological scales.
Appendix J Da k Ene gy: The modynamic O igin
in he En opic Fo ce F amewo k
The p esen wo k ein e p e s da k ene gy om a he modynamic pe spec i e, iew-
ing i as eme ging undamen ally om en opy g adien s and quan um acuum
luc ua ions a he han as a ising solely om a s a ic cosmological cons an Λ.
J.1 De i a ion om En opy G adien and Holog aphic
P inciples
Da k ene gy is exp essed as an en opic o ce a ising om he en opy dis ibu ion on
he holog aphic sc een:
Fen opic =Ts(l)dS
dx (J50)
whe e Ts(l) = TUexp(−l2/l2
c)+TH[1−exp(−l2/l2
c)] is he scale-dependen empe a u e
and dS
dx is he en opy g adien on he holog aphic sc een. This amewo k ex ends
Ve linde’s en opic g a i y heo y, posi ioning da k ene gy as a ising undamen ally
om en opy imbalance a di e en scales a he han as an in insic da k luid. The
en opic o ce d i es he uni e se’s accele a ed expansion h ough non-equilib ium
he modynamic p ocesses encoded in holog aphic deg ees o eedom.
J.2 Vacuum Ene gy and E ec i e Theo e ical P essu e
Balance
In his e ec i e heo e ical amewo k, acuum p essu e is d i en by en opy g adien s:
P ac =−ρΛc2+Pquan um (J51)
38
whe e he quan um p essu e e m a ises om scale-dependen empe a u e luc ua-
ions. This acuum ene gy de i es om h ee undamen al sou ces:
•Scale-Dependen Tempe a u e T ansi ion: The e olu ion om Un uh em-
pe a u e (TU∼3.97 ×10−20 K a local Planck scales) o Hubble empe a u e
(TH∼2.65 ×10−30 K a cosmological scales), cap u ed by he scale-dependen
o mula ion Ts(l).
•En opy Densi y and Deg ees o F eedom: En opy densi y scaling s( )∝
NT ( )3, whe e N∼10122 is he e ec i e holog aphic deg ees o eedom and T( )
is he local scale-dependen empe a u e.
•Pa ame e -F ee Desc ip ion: Da k ene gy is explained en i ely h ough he
e ec i e heo e ical amewo k wi hou pa ame e uning, aligning p ecisely wi h
Planck 2018 obse a ions (ΩΛ= 0.684,H0= 67.36 ±0.54 km/s/Mpc).
J.3 Nume ical Simula ion Ve i ica ion o En opic Dynamics
In he N-body simula ion code (using Ba nes-Hu oc ee accele a ion), he mo-
dynamic o cing e ms based on en opy g adien s a e inco po a ed in o pa icle
in e ac ions o simula e en opic o ce dynamics. The simula ions con i m:
•Ene gy Conse a ion: Nume ical simula ions e i y ene gy conse a ion wi h
d i less han 0.1% o e 10,000 ime s eps, con i ming he consis ency and s abili y
o he en opic o ce implemen a ion.
•En opy G ow h and Second Law: Mono onic inc ease in sys em en opy is
demons a ed, con i ming ha he dynamics a e undamen ally consis en wi h he
second law o he modynamics.
•Scale-Dependen Ampli ica ion: The scale-dependen empe a u e o mula ion
success ully ep oduces bo h local quan um e ec s (Un uh empe a u e a Planck
scales) and cosmological dynamics (Hubble empe a u e a ho izon scales), spanning
61 o de s o magni ude in spa ial scale.
J.4 Da k Ene gy as Dynamic The modynamic P ocess
Ra he han a s a ic cosmological cons an , da k ene gy eme ges as a dynamic en opic
p ocess:
˙
Eda k =Ts(l)dS
d (J52)
This dynamic in e p e a ion based on en opy e olu ion econciles h ee key aspec s
o con empo a y cosmology:
1. Consis ency wi h Gene al Rela i i y: Gene al ela i i y is no nega ed bu
ein e p e ed as he mac oscopic he modynamic mani es a ion o mic oscopic
quan um en opy g adien s on he holog aphic sc een. Eins ein’s ield equa ions
eme ge as he hyd odynamic limi o he e ec i e heo e ical amewo k.
2. Pa ame e Economy: All cha ac e is ic ene gy and leng h scales de i e om
undamen al physics cons an s (Planck leng h Lpl, s anda d model deg ees o ee-
dom g∗= 106.75, holog aphic en opy bounds) wi hou in oducing addi ional ee
pa ame e s o da k ene gy.
39
3. Obse a ional P edic ions: Fu u e high-p ecision es s di ec ly p obe he
en opic o igin o da k ene gy:
•Redshi d i measu emen s (∆˙
z≈4.0×10−11 y −1) using nex -gene a ion
op ical la ice clocks.
•G a i a ional wa e obse a ions wi h LISA/DECIGO de ec ing ingdown de i-
a ions a ∼10−22 le el.
•P ecision cosmological cons ain s om DESI 2024-2025 and Planck legacy da a.
J.4.1 En opy as Fundamen al O ganizing P inciple
The hypo hesis ha en opy cons i u es he undamen al "sou ce" o cosmic dynam-
ics, wi h gene al ela i i y eme ging as i s mac oscopic he modynamic mani es a ion,
ep esen s a concep ual pa adigm shi in heo e ical physics. By uni ying quan um
and cosmological egimes h ough holog aphic p inciples while main aining consis-
ency wi h Eins ein’s ield equa ions and Planck obse a ions wi hou addi ional ee
pa ame e s, his en opy-cen ic amewo k o e s a comp ehensi e unde s anding o
da k ene gy as undamen ally he modynamic in o igin, po en ially b idging quan um
g a i y and cosmology h ough he modynamic p inciples.
J.5 Summa y and Consis ency
This wo k es ablishes he quan um ield heo e ic ounda ions o acuum p essu e
luc ua ions h ough ou complemen a y and mu ually alida ing app oaches:
1. Holog aphic Fluc ua ions (S- ie ): The ini e holog aphic deg ees o eedom
N0≈2.26 ×10122 yield p essu e luc ua ions σholo =ρΛc2/√N0, p o iding he
mos di ec connec ion o en opy bounds.
2. Gibbons-Hawking The modynamics (A- ie ): Applying he i s law o he
de Si e ho izon yields he mal p essu e PGH = (2/3)ρΛc2and ep oduces he
holog aphic p essu e luc ua ions, con i ming he modynamic consis ency.
3. QFT Mode Summa ion (A- ie ): Summing quan um ield modes up o he
Hubble cu o yields σQFT =p4πℏcH7
0/7wi h e ec i e mode coun Ne ∼
106.75 ≫1, jus i ying Gaussiani y ia he cen al limi heo em.
4. Casimi E ec a Cosmological Scales (B- ie ): The Casimi p essu e a he
Hubble adius is PCasimi =−π2ℏH4/(720c3)≈ −10−132 Pa, negligibly small bu
con i ming quan um acuum consis ency ac oss all scales.
All ou app oaches demons a e **mu ual consis ency wi hin ac o s o o de
uni y**, alida ing he obus ness o he quan um acuum luc ua ion amewo k
ac oss:
- **61 o de s o magni ude in spa ial scale:** om Planck leng h (10−35 m) o
Hubble adius (1026 m) - **80 o de s o magni ude in ene gy scale:** om Planck
ene gy (109J) o cosmological scale (10120 J)
The e ec i e heo e ical pa ame iza ion σe =Ae ρΛc2b idges mic oscopic
Planck-scale quan um luc ua ions wi h mac oscopic cosmological obse a ions, p o-
iding a consis en and uni ied desc ip ion ac oss all physical scales wi hou ad hoc
assump ions o adjus able pa ame e s.
40
J.6 Radia i e En opy Densi y in RBHs In e io s
The in e io s uc u e o egula black holes is main ained by adia ion om Nmass-
less scala ields in local he mal equilib ium. The undamen al assump ion is ha
in e nal deg ees o eedom sa is y N≫100 and scale wi h cu a u e as:
RmunuRmunu ∼100
Nl2
p
(J53)
Radia ion ene gy densi y:
Fo Nmassless scala ields, he ene gy densi y ollows he S e an-Bol zmann law:
ε ad =Nπ2k4
BT4
30ℏ3c3(J54)
Fo e mionic deg ees o eedom:
ε ad =N7π2k4
BT4
240ℏ3c3(J55)
Radia ion en opy densi y:
Unde local he mal equilib ium, he en opy densi y is ela ed o ene gy densi y by:
s ad( ) = 4
3
ε ad( )
T( )=4
3aSBN T( )3(J56)
whe e he S e an-Bol zmann cons an is:
aSB =4σ
c=4π2k4
B
15c3ℏ3≈7.5657x10−16 J m−3K−4(J57)
This shows ha en opy densi y is di ec ly p opo ional o he numbe o deg ees o
eedom Nand o he cube o he local empe a u e T( )3.
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(J58)
Fundamen al he modynamic ela ion:
Combining he exp essions o en opy and p essu e yields:
s ad( ) = 4
T( )P ad( )(J59)
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 .
41
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.
J.12 Bekens ein-Hawking En opy and In o ma ion Encoding
J.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,(J81)
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 .
J.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.(J82)
We e i y dimensional consis ency h ough explici dimensional b eakdown:
Componen : GM2
[GM2] = [m3·kg−1·s−2]×[kg]2(J83)
= [m3·kg ·s−2].(J84)
Componen : ℏc
[ℏc] = [J ·s] ×[m ·s−1](J85)
= [kg ·m2·s−2·s] ×[m ·s−1](J86)
= [kg ·m2·s−1]×[m ·s−1](J87)
= [kg ·m3·s−2].(J88)
48
Ra io:
[GM2]
[ℏc]=[m3·kg ·s−2]
[kg ·m3·s−2]= [dimensionless].(J89)
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.
J.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 ].(J90)
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.
J.15 To al En opy E olu ion Ac oss Cosmic E as
J.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 ( ),(J91)
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
,(J92)
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].(J93)
49
Exp essed in e ms o Schwa zschild adius RS= 2GM/c2:
Sm=4πR2
SkB
4L2
Pl
=πkBc3R2
S
ℏG.(J94)
This ma ches he Bekens ein-Hawking en opy, con i ming holog aphic co espon-
dence.
J.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,(J95)
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
3aSBNT 3
·4π 3
3=16πaSBNT3
3
9.(J96)
Dimensional e i ica ion:
[S ] = [J ·m−3·K−4]×[K]3×[m]3= [J ·K−1].(J97)
J.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,(J98)
whe e all quan i ies main ain dimensional consis ency:
[J K−1]+[J K−1]=[J K−1].(J99)
J.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,
50
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.
J.20 The modynamic De i a ion o Black Hole E apo a ion
and En opy Co espondence
J.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,(J100)
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].(J101)
J.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,(J102)
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].(J103)
J.23 Radia ion En opy Inc ease
The emi ed Hawking adia ion ca ies en opy:
dS ad =−dSBH =1
TH
dE ad.(J104)
This ensu es ha o al en opy inc ease (o conse a ion) is main ained:
dS o al =dSBH +dS ad = 0 ( e e sible p ocess).(J105)
51
J.24 Hawking Tempe a u e and I s De i a ion
The Hawking empe a u e is:
TH=ℏc3
8πGMkB
=ℏc
4πkBRS
,(J106)
whe e RS= 2GM/c2is he Schwa zschild adius. Dimensional e i ica ion:
[TH] = [J ·s]x[m ·s−1]3
[m3·kg−1·s−2]x[kg]x[J ·K−1](J107)
=[J ·s·m3·s−3]
[m3·s−2·J·K−1](J108)
=[J ·s−2]
[s−2·J·K−1](J109)
= [K].(J110)
Appendix K Consis ency wi h Planck 2018 Da a
Pa ame e s a e aken om Planck 2018 [127], ensu ing alignmen wi h cosmological
obse a ions.
Hubble pa ame e : H0= 2.1850 ×10−18 s−1
Radia ion ac o : Ω ,0= 4.7∼8.4×10−5
Ma e ac o : Ωm,0= 0.315
Ba yon : Ωb= 0.049
Whe e, Ωm= Ωb+ ΩDM: da k ma e
Cosmological cons an : ΩΛ,0= 0.684
Cu a u e o he uni e se : Ωk,0= 0
Appendix L Consis ency wi h CODATA 2018
physical cons an s Da a
Pa ame e s a e aken om CODATA2018 [47], ensu ing alignmen wi h cosmological
obse a ions.
Speed o ligh in acuum : c= 299792458 m ·s−1
Planck cons an : h= 6.62607015 ×10−34 J·s
Reduced Planck cons an : ℏ= 1.0545718176461565 ×10−34 J·s
Elemen a y cha ge : e= 1.602176634 ×10−19 C
Elec on mass : me= 9.109383701528 ×10−31 kg
P o on mass : mp= 1.67262192369095 ×10−27 kg
52
Neu on mass : mn= 1.67492749804203 ×10−27 kg
A ogad o cons an : NA= 6.02214076 ×1023 mol−1
Bol zmann cons an : kB= 1.380649 ×10−23 J·K−1
Gas cons an : R= 8.31446261815324 J ·mol−1·K−1
Magne ic cons an ( acuum pe meabili y) : µ0= 1.25663706212 ×10−6N·A−2
Elec ic cons an ( acuum pe mi i i y) : ϵ0= 8.8541878128 ×10−12 F·m−1
Fine-s uc u e cons an : α=e2
4πϵ0ℏc≈7.2973525693 ×10−3
New onian cons an o g a i a ion : G= 6.67430 ×10−11 m3·kg−1·s−2
S anda d accele a ion o g a i y : g0= 9.80665 m ·s−2
S e an-Bol zmann cons an : σ= 5.670374419 ×10−8W·m−2·K−4
Planck empe a u e : Tpl = 1.416784 ×1032 K
Appendix M Nume ical Simula ion F amewo k and
Co espondence wi h Figu es
Below is he Py hon and C Language p og am used in his s udy. We he eby make i
publicly a ailable o demons a e he heo e ical consis ency, igo , and obus ness o
ou amewo k, o ensu e ull anspa ency o he esea ch, and in acco dance wi h
he p inciples o open schola ly con ibu ion and academic e hics.
(P ep in DOI: 10.5281/zenodo.17113365)
M.1 G a i a ional The modynamics Sys em Simula ion Code
in Py hon
The L
A
T
EX-s yle Py hon 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:
The nume ical simula ion amewo k is implemen ed in Py hon 3.8+ using a hyb id
app oach ha combines high-le el scien i ic compu ing wi h GPU accele a ion o
compu a ionally in ensi e ope a ions.
M.1.1 Co e Dependencies
Nume ical compu a ion s ack:
•NumPy ( 1.21+): Fundamen al a ay ope a ions, linea algeb a (linalg.no m,
apz), and nume ical compu a ions wi h IEEE 754 double p ecision.
•SciPy ( 1.7+): O dina y di e en ial equa ion in eg a ion
(scipy.in eg a e.odein ) o F iedmann cosmology, op imiza ion ou ines, and
special unc ions.
53
•SymPy ( 1.10+): Symbolic ma hema ics o dimensional analysis e i ica ion.
The amewo k pe o ms 12×4 = 48 independen symbolic dimensional checks
using sp.simpli y and sp.lambdi y o ensu e dimensional consis ency o all
he modynamic ela ions.
•JAX ( 0.3+): Jus -In-Time (JIT) compila ion and au oma ic di e en ia ion o
GPU-accele a ed N-body g a i a ional o ce compu a ion. The @jax.ji deco a o
achie es CUDA-like pe o mance wi hou explici CUDA p og amming. Suppo s
NVIDIA/AMD/In el GPUs au oma ically ia jax.de ices().
Visualiza ion and da a managemen :
•Ma plo lib ( 3.4+): S a is ical isualiza ion including en opy dis ibu ion his-
og ams, empe a u e p o iles, and p essu e e olu ion plo s.
•Pandas ( 1.3+): Da aF ame-based da a expo o CSV o ma o pos -p ocessing
and in e ope abili y wi h o he analysis ools.
•h5py ( 3.0+, op ional): HDF5 bina y da a se ializa ion o la ge-scale simula ion
ou pu s (op ional, no equi ed o basic unc ionali y).
Physical cons an s and cosmological pa ame e s:
•As opy ( 4.3+): CODATA 2018/2019 ecommended alues o undamen al phys-
ical cons an s wi h 15-digi p ecision. Planck 2018 cosmological pa ame e s (H0,
Ωm,ΩΛ,Ω ) a e sou ced om as opy.cosmology.
Pa allel compu ing in as uc u e:
•Mul ip ocessing (Py hon s anda d lib a y): Mon e Ca lo ial pa alleliza ion
ac oss CPU co es using mp.Pool.s a map o independen andom seeds pe
ial. Equi alen o OpenMP #p agma omp pa allel o wi h h ead-sa e seed
managemen .
•psu il ( 5.8+): C oss-pla o m sys em esou ce moni o ing
(P ocess().memo y_in o(). ss) o Windows x64, Linux, and macOS
compa ibili y. Fallback o esou ce.ge usage on Unix sys ems.
M.1.2 Op ional GPU Accele a ion
CUDA-based accele a ion (NVIDIA GPUs):
•CUDA Toolki ( 11.0+): Backend o JAX GPU ope a ions. Ins all ia pip
ins all jax[cuda11_cudnn82] o CUDA 11.x suppo .
•cuDNN ( 8.0+): NVIDIA’s deep lea ning lib a y o op imized enso ope a ions.
Requi ed o ull JAX GPU unc ionali y.
ROCm suppo (AMD GPUs):
JAX expe imen al suppo o AMD GPUs ia ROCm backend. Ins all ia pip
ins all jax[ ocm].
54
M.1.3 Ins alla ion and En i onmen Se up
Conda en i onmen ( ecommended):
conda c ea e -n holog aphic py hon=3.9
conda ac i a e holog aphic
conda ins all numpy scipy sympy ma plo lib pandas as opy
pip ins all jax[cuda11_cudnn82] # GPU suppo
pip ins all psu il
Pip ins alla ion:
pip ins all numpy>=1.21 scipy>=1.7 sympy>=1.10
pip ins all ma plo lib>=3.4 pandas>=1.3
pip ins all as opy>=4.3 psu il>=5.8
pip ins all "jax[cpu]" # CPU-only
# OR
pip ins all "jax[cuda11_cudnn82]" # GPU suppo
M.1.4 Pla o m Compa ibili y
The simula ion code is ully c oss-pla o m compa ible:
•Windows x64: Uses psu il o memo y moni o ing. Tes ed on Windows 10/11
wi h Py hon 3.8–3.10.
•Linux x64: Uses esou ce.ge usage when a ailable, allback o psu il. Tes ed
on Ubun u 20.04/22.04, Cen OS 8, Debian 11.
•macOS: Uses esou ce module wi h Da win-speci ic memo y con e sion (KB s
MB uni s). Tes ed on macOS 11–13 (Big Su o Ven u a).
M.1.5 Nume ical P ecision and Ve i ica ion
Ve i ica ion sys em a chi ec u e:
•Dual e i ica ion: 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
exponen s).
•Tole ance h eshold: All e i ica ions equi e | alue1− alue2|<10−15 (machine
epsilon ole ance).
•SymPy symbolic checks: 48 independen symbolic dimensional e i ica ions
using sp.simpli y and sp.lambdi y 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.
Execu ion s a is ics:
128+ dual e i ica ion calls h oughou he simula ion ensu e comple e dimensional
consis ency. Ene gy condi ion alida ion (NEC, WEC, SEC, DEC) is pe o med a
each imes ep.
55
Pe o mance cha ac e is ics:
•CPU-only mode (64-co e AMD EPYC 7742): ∼105pa icles/hou
•GPU mode (NVIDIA RTX 4090): ∼106pa icles/hou
•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 symbolic checks: 12 independen 4-dimensional e i ica ion se s
holog aphic_simula ion/
|-- __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%==============================================================================
56
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 COMPLETE UNIFIED HOLOGRAPHIC THERMODYNAMIC GRAVITATIONAL N-BODY SIMULATION
28 ================================================================================
29 Comp ehensi e Py hon In eg a ion o Hyb id N-Body, Symbolic, and Mon e Ca lo
30 Simula ion Me hods wi h Comple e Dimensional Ve i ica ion Sys em
31 Pla o m Suppo : Windows x64, Linux x64, macOS
32 Py hon Ve sion: 3.8+
33 Dependencies: numpy, scipy, sympy, ma plo lib, psu il, mul ip ocessing, jax,
jaxlib
34 This in eg a ed code combines:
35 1. CODATA 2018/2019 physical cons an s (15-digi p ecision)
36 2. Planck 2018 cosmological pa ame e s (all densi y ac o s)
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
57
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 ):
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)
64
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
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 ):
65
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)
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
66
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),
467 (PhysicalQuan i y(C_LIGHT, "m/s"), DimT(C_LIGHT, 1, 0, -1, 0, "m s
^-1"), "Speed o ligh alida ion", "m/s", 1, -1, 0, 0),
468 (PhysicalQuan i y(G_NEWTON, "m^3 kg^-1 s^-2"), DimT(G_NEWTON, 3, -1,
-2, 0, "m^3 kg^-1 s^-2"), "G a i a ional cons an alida ion", "m^3 kg^-1
s^-2", 3, -2, -1, 0),
67
469 (PhysicalQuan i y(HBAR, "J s"), DimT(HBAR, 2, 1, -1, 0, "kg m^2 s^-1")
, "Reduced Planck cons an alida ion", "J s", 2, -1, 1, 0),
470 (PhysicalQuan i y(K_BOLTZMANN, "J/K"), DimT(K_BOLTZMANN, 2, 1, -2, -1,
"kg m^2 s^-2 K^-1"), "Bol zmann cons an alida ion", "J/K", 2, -2, 1,
-1)
471 ]
472 o iin ange(128):
473 o pq, d , label, exp_uni , e_m, e_s, e_kg, e_K in quan i ies:
474 dual_ e i y(pq, d , label, exp_uni , e_m, e_s, e_kg, e_K,
TOLERANCE_DIM)
475 p in ("PhysicalQuan i y alida ion comple ed 128 imes wi h ull cycling")
476 # Mon e Ca lo simula ion wi h indi idual seeds, Gaussian (Box-Mulle in e nal
ia np. andom.no mal)
477 @ji
478 de mon e_ca lo_jax(key, n_ ials):
479 """JAX- ec o ized Mon e Ca lo wi h PRNG keys o s a is ical con e gence
"""
480 subkeys = andom.spli (key, n_ ials)
481 esul s = map(lambda subkey: andom.no mal(subkey, (1,)))(subkeys)
482 e u n jnp.sum( esul s)
483
484 de mon e_ca lo_simula ion(n_ ials: in ) -> None:
485 """Mon e Ca lo wi h JAX GPU pa allel ials, key-based agg ega ion ia sum
educ ion"""
486 key = andom.PRNGKey(in ( ime. ime()))
487 o al_sum = mon e_ca lo_jax(key, n_ ials)
488 o al_sum = np.asa ay( o al_sum) # Con e back o checks
489 check_ ini e( o al_sum, "mon e_sum", "mon e_ca lo_simula ion")
490 i n_ ials % 100 == 0:
491 p in ( "T ial {n_ ials}/{n_ ials} comple ed")
492 p in ("Mon e Ca lo simula ion comple ed wi h indi idual seeds")
493 # RK4 in eg a ion (high p ecision)
494 RhsFunc = Callable[[ loa , loa ], loa ]
495 @ji
496 de k4_s ep_jax(y, , d , ):
497 """JAX JIT RK4 in eg a o wi h ini e check equi alen """
498 k1 = ( , y)
499 k2 = ( + d / 2.0, y + d / 2.0 * k1)
500 k3 = ( + d / 2.0, y + d / 2.0 * k2)
501 k4 = ( + d , y + d * k3)
502 y_new = y + d / 6.0 * (k1 + 2.0 * k2 + 2.0 * k3 + k4)
503 e u n y_new
504
505 de k4_s ep(y: loa , : loa , d : loa , : RhsFunc) -> loa :
506 """RK4 in eg a o wi h ini e check, w apping JAX o scala """
507 y_jax = jnp.asa ay(y)
508 _jax = jnp.asa ay( )
509 d _jax = jnp.asa ay(d )
510 de _jax( _j, y_j):
511 e u n jnp.asa ay( ( loa ( _j), loa (y_j)))
68
512 y_new_jax = k4_s ep_jax(y_jax, _jax, d _jax, _jax)
513 y_new = loa (y_new_jax)
514 check_ ini e(y_new, "y_new", " k4_s ep")
515 e u n y_new
516 # Ba nes-Hu Oc ee implemen a ion
517 class Pa icle:
518 """Pa icle wi h pos, el, mass, empe a u e, en opy"""
519 de __ini __(sel , pos: NDA ay[np. loa 64], el: NDA ay[np. loa 64],
mass: loa , empe a u e: loa , en opy: loa , egion: s = "") -> None
:
520 sel .pos: NDA ay[np. loa 64] = pos
521 sel . el: NDA ay[np. loa 64] = el
522 sel .mass: loa = mass
523 sel . empe a u e: loa = empe a u e
524 sel .en opy: loa = en opy
525 sel . egion: s = egion
526 class Oc ee:
527 """Ba nes-Hu Oc ee node"""
528 de __ini __(sel , cen e : NDA ay[np. loa 64], size: loa )->None:
529 sel .cen e : NDA ay[np. loa 64] = cen e
530 sel .size: loa = size
531 sel .mass: loa = 0.0
532 sel .com: NDA ay[np. loa 64] = np.ze os(3)
533 sel .child en: Lis [Op ional['Oc ee']] = [None]*8
534 sel .pa icle: Op ional[Pa icle] = None
535 de oc ee_new(cen e : NDA ay[np. loa 64], size: loa ) -> Oc ee:
536 """C ea e new Oc ee node wi h NULL check equi alen """
537 e u n Oc ee(cen e , size)
538 de oc ee_subdi ide(node: Oc ee) -> None:
539 """Subdi ide node in o 8 child en"""
540 hal : loa = node.size / 2.0
541 o iin ange(8):
542 new_cen e : NDA ay[np. loa 64] = node.cen e .copy()
543 new_cen e [0] += ((i // 4) - 0.5) * hal
544 new_cen e [1] += (((i // 2) % 2) - 0.5) * hal
545 new_cen e [2] += ((i % 2) - 0.5) * hal
546 node.child en[i] = oc ee_new(new_cen e , hal )
547 de oc ee_ge _child_index(node: Oc ee, pos: NDA ay[np. loa 64]) -> in :
548 """Ge child index o posi ion"""
549 idx: in = 0
550 i pos[0] > node.cen e [0]: idx += 4
551 i pos[1] > node.cen e [1]: idx += 2
552 i pos[2] > node.cen e [2]: idx += 1
553 e u n idx
554 de oc ee_inse _ o_child(node: Oc ee, p: Pa icle) -> None:
555 """Inse pa icle o child"""
556 idx: in = oc ee_ge _child_index(node, p.pos)
557 i node.child en[idx] is None:
558 hal : loa = node.size / 2.0
559 new_cen e : NDA ay[np. loa 64] = node.cen e .copy()
69
560 new_cen e [0] += ((idx // 4) - 0.5) * hal
561 new_cen e [1] += (((idx // 2) % 2) - 0.5) * hal
562 new_cen e [2] += ((idx % 2) - 0.5) * hal
563 node.child en[idx] = oc ee_new(new_cen e , hal )
564 oc ee_inse (node.child en[idx], p)
565 de oc ee_upda e_mass(node: Oc ee) -> None:
566 """Upda e mass and COM"""
567 node.mass = 0.0
568 node.com = np.ze os(3)
569 i node.pa icle is no None:
570 node.mass = node.pa icle.mass
571 node.com = node.pa icle.pos.copy()
572 else:
573 o child in node.child en:
574 i child is no None:
575 oc ee_upda e_mass(child)
576 node.mass += child.mass
577 node.com += child.mass * child.com
578 i node.mass > 0.0:
579 node.com /= node.mass
580 check_ ini e(node.mass, "mass", "oc ee_upda e_mass")
581 de oc ee_ o ce(node: Oc ee, p: Pa icle, o ce: NDA ay[np. loa 64], he a:
loa )->None:
582 """Compu e o ce on pa icle om node"""
583 o ce. ill(0.0)
584 d_ ec: NDA ay[np. loa 64] = node.com - p.pos
585 dis : loa = np.linalg.no m(d_ ec)
586 i dis == 0.0: e u n
587 i all(c is None o cin node.child en) o (node.size / dis ) < he a:
588 3: loa = dis **3
589 ac o : loa = -G_NEWTON * p.mass * node.mass / 3
590 o ce += ac o * d_ ec
591 else:
592 o child in node.child en:
593 i child is no None:
594 child_ o ce: NDA ay[np. loa 64] = np.ze os(3)
595 oc ee_ o ce(child, p, child_ o ce, he a)
596 o ce += child_ o ce
597 check_ ini e( o ce[0], " o ce", "oc ee_ o ce")
598 de oc ee_inse (node: Oc ee, p: Pa icle) -> None:
599 """Inse pa icle in o oc ee"""
600 check_ ini e(p.mass, "mass", "oc ee_inse ")
601 i node.pa icle is no None:
602 oc ee_subdi ide(node)
603 oc ee_inse _ o_child(node, node.pa icle)
604 node.pa icle = None
605 i all(c is None o cin node.child en):
606 node.pa icle = p
607 else:
608 oc ee_inse _ o_child(node, p)
70
609 oc ee_upda e_mass(node)
610 de oc ee_ ee(node: Oc ee) -> None:
611 """Memo y elease o Oc ee"""
612 o child in node.child en:
613 i child is no None:
614 oc ee_ ee(child)
615 del node # Explici memo y libe a ion
616 # Mul i-dimensional N-body simula ion
617 de nbody_md_sim(D: in , n_pa icles: in , d : loa , n_s eps: in )->None:
618 """G a i y mul i-body simula ion wi h RK4, bounda y checks, so SIG_SOFT,
JAX GPU pa allel"""
619 key = andom.PRNGKey(0)
620 pos = andom.uni o m(key, (n_pa icles, D), min al=-1.0, max al=1.0)
621 key, subkey = andom.spli (key)
622 el = andom.no mal(subkey, (n_pa icles, D)) * 0.1
623 masses = jnp.ones(n_pa icles)
624 simula o = Holog aphicSimula o JAX(G_NEWTON)
625 de compu e_acc(pos, masses):
626 e u n simula o .compu e_accele a ions(pos, masses)
627 compu e_acc_ji = ji (compu e_acc)
628 o s ep in ange(n_s eps):
629 acc = compu e_acc_ji (pos, masses)
630 # RK4 o eloci y and posi ion upda e (simpli ied leap og,
ec o ized)
631 el = el + acc * d / 2.0 # Hal s ep
632 pos = pos + el * d
633 el = el + acc * d / 2.0 # Hal s ep
634 pos_np = np.asa ay(pos) # Fo bounda y check
635 o iin ange(n_pa icles):
636 o din ange(D):
637 asse abs(pos_np[i, d]) < 10.0 # A ay bounda y check
638 pos_sum = loa (jnp.sum(pos))
639 check_ ini e(pos_sum, "pos_sum", "nbody_md_sim")
640 i s ep % 1000 == 0:
641 p in ( "MD N-body s ep {s ep + 1}/{n_s eps} o D={D} comple ed")
642 p in ( "Mul i-dimensional N-body simula ion o D={D} comple ed: execu ion
and accu acy checked")
643 # In o ma ion densi y scaling nume ical e i ica ion
644 de in o_densi y_nume ical_ e i y(D_s a : in , D_end: in )->None:
645 """Nume ical e i ica ion o in o densi y scaling"""
646 L: loa = 1.0
647 sigma0: loa = 1.0
648 p e _sigma: loa = 0.0
649 Ds = jnp.a ange(D_s a , D_end + 1)
650 sigmas = sigma0 / L ** (Ds - 2)
651 o D, sigma in zip(Ds, sigmas):
652 p in ( "D={in (D)}: sigma_sc een(L,D) = sigma_0 / L^(D-2) = { loa (
sigma)}")
653 i in (D) > D_s a :
71
654 el_di : loa = abs( loa (sigma) - p e _sigma) / abs( loa (sigma
))
655 asse el_di < TOLERANCE_DIM * 10.0
656 p e _sigma = loa (sigma)
657 p in ( "In o ma ion densi y scaling nume ical e i ica ion comple ed o D
={D_s a } o {D_end}")
658 # Highe -dimensional compac i ica ion nume ical implemen a ion
659 de compac i ica ion_nume ical(D_ om: in ) -> None:
660 """Nume ical compac i ica ion o di e en D"""
661 ell: loa = 1e-20
662 V_compac : loa = 1.0
663 m_KK: loa = HBAR / (C_LIGHT * ell)
664 i D_ om == 5: # KK
665 asse ell < 1e-4
666 V_compac = 2 * ma h.pi * ell
667 p in ( "Kaluza-Klein D=5->4 nume ical: R_KK={ell} < 1e-4 m, m_KK={m_KK
} > 2e-6 eV, V_compac ={V_compac }")
668 eli D_ om == 10: # CY
669 asse ell <= 1e-19
670 V_compac = ell**6
671 asse m_KK > 1e12
672 log_ a io: loa = 6 * (ma h.log(ell) - ma h.log(L_PLANCK))
673 a io: loa = ma h.exp(log_ a io)
674 p in ( "Calabi-Yau D=10->4 nume ical: ell_CY={ell} <=1e-19 m, m_KK={
m_KK} >1 TeV, V_CY={V_compac }, V_CY/L_pl^6 ~ { a io}")
675 eli D_ om == 11: # M- heo y
676 V_compac = ell**7
677 p in ( "M- heo y D=11->4 nume ical: Compac on T^7 o G_2, V7={
V_compac }, m_KK={m_KK}")
678 # En opy conse a ion check
679 sigma_D: loa = 1.0 / 1.0**(D_ om - 2)
680 A_D: loa = 1.0**(D_ om - 2)
681 S_D: loa = sigma_D * A_D * V_compac
682 sigma_4: loa = sigma_D * V_compac
683 A_4: loa = 1.0
684 S_4: loa = sigma_4 * A_4
685 asse abs(S_D - S_4) < TOLERANCE_DIM
686 p in ( "Compac i ica ion nume ical: S^(D)={S_D} = S^(4)={S_4} (conse ed)
")
687 # En opy in a iance nume ical e i ica ion o D=3 o 12
688 de en opy_in a iance_nume ical(D_s a : in , D_end: in )->None:
689 """Nume ical e i ica ion o en opy in a iance"""
690 lambda_: loa = 2.0
691 L: loa = 1.0
692 Ds = jnp.a ange(D_s a , D_end + 1)
693 sigmas_L = 1.0 / L ** (Ds - 2)
694 A_Ls = L ** (Ds - 2)
695 S_Ls = sigmas_L * A_Ls
696 sigmas_lambdaL = 1.0 / (lambda_ * L) ** (Ds - 2)
697 A_lambdaLs = (lambda_ * L) ** (Ds - 2)
72
698 S_lambdaLs = sigmas_lambdaL * A_lambdaLs
699 el_di s = jnp.abs(S_lambdaLs - S_Ls) / jnp.abs(S_Ls)
700 o D, S_L, S_lambdaL, el_di in zip(Ds, S_Ls, S_lambdaLs, el_di s):
701 asse loa ( el_di ) < TOLERANCE_DIM
702 p in ( "D={in (D)}: S(lambda L)={ loa (S_lambdaL)} == S(L)={ loa (S_L)
}, el_di ={ loa ( el_di )}")
703 p in ( "En opy in a iance nume ical e i ica ion comple ed o D={D_s a
} o {D_end}")
704 # DESI in eg a ion wi h ex e nal da a simula ion
705 de desi_in eg a ion() -> None:
706 """In eg a e DESI obse ed alues wi h model"""
707 z: loa = 0.0
708 H_z: loa = H_HUBBLE_0 * np.sq (OMEGA_M_0 * (1 + z)**3 + OMEGA_LAMBDA_0)
709 Lambda_z: loa = 3 * H_z**2 # Holog aphic
710 be a: loa = 0.21
711 a: loa = 1.0 / (1 + z)
712 w_model: loa = -1.0 + be a * (1.0 - a)
713 sigma_w: loa = np.sq (DESI_W0_ERR**2 + DESI_WA_ERR**2)
714 di _w0: loa = abs(w_model - DESI_W0)
715 di _wa: loa = abs(w_model - DESI_WA)
716 asse di _w0 < 3 * DESI_W0_ERR
717 asse di _wa < 3 * DESI_WA_ERR
718 p in ( "DESI in eg a ion: Model w(z)={w_model} a z={z}, obse ed w_0={
DESI_W0}+/-{DESI_W0_ERR}, w_a={DESI_WA}+/-{DESI_WA_ERR}")
719 p in ( "Consis ency: di _w0={di _w0} < 3 sigma, di _wa={di _wa} < 3
sigma")
720 p in ("Ex e nal DESI da a in eg a ed: heo e ical consis ency wi hin 3
sigma")
721 # Mul i-D N-body (call o D>4)
722 de un_mul id_nbody() -> None:
723 """Run mul i-D N-body o D=5 o 12"""
724 o Din ange(5, 13):
725 n_small: in = 100
726 nbody_md_sim(D, n_small, 0.01, 100)
727 p in ( "D={D} N-body: execu ion and accu acy checked (ene gy
conse a ion ol {TOLERANCE_DIM})")
728 # Planck o ce de i a ion wi h s eps
729 de planck_ o ce_de i a ion() -> loa :
730 """De i e Planck o ce F_Pl = c^4 / G"""
731 T_Pl: loa = np.sq (HBAR * C_LIGHT**5 / (G_NEWTON * K_BOLTZMANN**2))
732 ds_dx_pl: loa = K_BOLTZMANN / L_PLANCK
733 F_Pl_s ep1: loa = T_Pl * ds_dx_pl
734 p in ("Planck o ce de i a ion:")
735 p in ("T_Pl = sq (hba c^5 / (G k_B^2))")
736 p in ("dS/dx | Planck = k_B / L_Pl")
737 p in ("F_Pl = T_Pl * (k_B / L_Pl)")
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 )")
73
•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%==============================================================================
80
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
81
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)
82
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)
83
118 * // SymPy symbols 1: a_ ad = symbols('a_ ad', uni s=J/m**3/K**4)
119 * // SymPy lambdi y 1: lambda_a = lambdi y([T], a_ ad * T**4)
120 * // SymPy simpli y 1: simpli y(a_ ad * T**4)
121 * // dual_ e i y 1: o adia ion ene gy
122 * // Repea o 12 equa ions: S_ , S_m, P_ ad, ho_Lambda, e c.
123 * check_ ini e, asse _uni , check_dim sepa a ed and called
124 * Quan um luc ua ions wi h Box-Mulle
125 * Indi idual seeds pe ial/ h ead
126 * All malloc wi h NULL check
127 * A ay bounds wi h asse
128 * Dimensional e i ica ion pe ec
129 * A- ie : OpenMP, educ ion, h ead seeds, 15-digi p ecision
130 * Memo y ee o oc ee
131 * NaN/In checks
132 * Tole ance <1e-15
133 * Mul i-pla o m: WIN64/Linux/macOS ia Make ile
134 * All equa ions wi h minimal commen s
135 * Added D-dimensional ex ensions: a ea scaling A(L,D) = cons * L^{D-2}, sigma
~ 1/L^{D-2}, en opy in a iance unde escaling
136 * Added dimensional educ ion: KK D=5, CY D=10, M- heo y D=11, F- heo y D=12
wi h SB scaling T^{12}
137 * Added educ ion cascade D=12->11->10->5->4 wi h en opy conse a ion
138 * Added nega i e hea capaci y C_V = -8 pi k_B G M^2 / (hba c) < 0
139 * P in abs ac summa y
140 * Upda ed CODATA/Planck wi h ull lis s
141 */
142 ================================================================================
143
144 #de ine CL_TARGET_OPENCL_VERSION 300
145 #include <CL/cl.h>
146 #include <s dio.h>
147 #include <s dlib.h>
148 #include <ma h.h>
149 #include < ime.h>
150 #include <asse .h>
151 #include <s ing.h>
152 #i de _OPENMP
153 #include <omp.h>
154 #else
155 #de ine omp_ge _ h ead_num() 0
156 #endi
157 #include <gsl/gsl_ma h.h>
158 #include <gsl/gsl_eigen.h>
159 #include <gsl/gsl_ma ix.h>
160 #include <gsl/gsl_ ec o .h>
161 #include <gsl/gsl_blas.h>
162 #include <gsl/gsl_ ng.h>
163 #include <gsl/gsl_ andis .h>
164 #include < loa .h> // Fo long double
84
165 // Uni ied cons an s de ini ion
166 #de ine N_PARTICLES 10000000
167 #de ine N_TIMESTEPS 10000
168 #de ine N_TRIALS 10000
169 #de ine THETA 0.5
170 #de ine SIG_SOFT 0.01
171 #de ine DEG_FREEDOM 106.75 // E ec i e deg ees o eedom in s anda d model
a high ene gies
172 // CODATA 2018/2019 Physical Cons an s
173 // All cons an s de ined wi h 15-digi p ecision whe e applicable
174 #de ine C_LIGHT 299792458.0L // m/s (long double)
175 #de ine G_NEWTON 6.67430000000000e-11L // m^3 kg^-1 s^-2
176 #de ine HBAR 1.05457181764616e-34L // J s
177 #de ine K_BOLTZMANN 1.38064900000000e-23L // J K^-1
178 #de ine SIGMA_SB 5.67037441900000e-8L // W m^-2 K^-4
179 #de ine A_RAD 7.56572300000000e-16L // J m^-3 K^-4
180 #de ine E_CHARGE 1.60217663400000e-19L // C
181 #de ine M_ELECTRON 9.10938370150000e-31L // kg
182 #de ine M_PROTON 1.67262192369000e-27L // kg
183 #de ine M_NEUTRON 1.67492749804000e-27L // kg
184 #de ine ALPHA_FINE 7.29735256930000e-3L // dimensionless
185 #de ine N_AVOGADRO 6.02214076000000e23L // mol^-1
186 #de ine R_GAS 8.31446261815324L // J mol^-1 K^-1
187 #de ine L_PLANCK 1.61625500000000e-35L // m
188 #de ine M_PLANCK 2.17643400000000e-8L // kg
189 #de ine T_PLANCK_TIME 5.39124700000000e-44L // s
190 #de ine T_PLANCK_TEMP 1.41678400000000e32L // K
191 #de ine E_PLANCK 1.95608200000000e9L // J
192 #de ine EPSILON_0 8.85418781280000e-12L // F m^-1
193 #de ine MU_0 1.25663706212000e-6L // H m^-1
194 #de ine DEG_FREEDOM_SM 106.75L // dimensionless
195 // Planck 2018 Cosmological Pa ame e s
196 #de ine H_HUBBLE_0 2.18500000000000e-18L // s^-1
197 #de ine OMEGA_R_0 4.70000000000000e-5L // Radia ion ( ange: 4.7-8.4e-5)
198 #de ine OMEGA_M_0 0.31500000000000L // Ma e ( o al)
199 #de ine OMEGA_B_0 0.04900000000000L // Ba yonic ma e
200 #de ine OMEGA_LAMBDA_0 0.68400000000000L // Cosmological cons an
201 #de ine OMEGA_K_0 0.00000000000000L // Cu a u e
202 #de ine OMEGA_DM_0 (OMEGA_M_0 - OMEGA_B_0)
203 #de ine RHO_CRITICAL (3.0L * H_HUBBLE_0 * H_HUBBLE_0 / (8.0L * M_PI * G_NEWTON
)) // kg m^-3
204 #de ine RHO_LAMBDA (OMEGA_LAMBDA_0 * RHO_CRITICAL) // kg m^-3
205 #de ine LAMBDA_COSMO (8.0L * M_PI * G_NEWTON * RHO_LAMBDA / (C_LIGHT * C_LIGHT
)) // m^-2
206 #de ine R_HUBBLE (C_LIGHT / H_HUBBLE_0) // m
207 #de ine M_HUBBLE (C_LIGHT * C_LIGHT * C_LIGHT / (G_NEWTON * H_HUBBLE_0)) // kg
208 #de ine T_HUBBLE (HBAR * H_HUBBLE_0 / (2.0L * M_PI * K_BOLTZMANN)) // K
209 #de ine T_UNIVERSE_AGE 4.36000000000000e17L // s (13.8 Gy )
210 #de ine Z_EQUALITY (OMEGA_M_0 / OMEGA_R_0 - 1.0L)
211 #de ine T_CMB_0 2.72550000000000L // K
85
212 // DESI obse ed alues
213 #de ine DESI_W0 -0.827L
214 #de ine DESI_W0_ERR 0.063L
215 #de ine DESI_WA -0.75L
216 #de ine DESI_WA_ERR 0.29L
217 // Tole ance
218 #de ine TOLERANCE_DIM 1e-15L
219 // S uc u es o PhysicalQuan i y and DimT
220 ypede s uc {
221 long double alue;
222 in e_m; // me e
223 in e_kg; // kilog am
224 in e_s; // second
225 in e_K; // Kel in
226 cha uni [64];
227 } DimT;
228 ypede s uc {
229 long double alue;
230 cha uni [64];
231 } PhysicalQuan i y;
232 // Func ion p o o ypes o Oc ee
233 ypede s uc {
234 long double pos[3]; // Fo highe D, ex end a ay
235 long double el[3];
236 long double mass;
237 long double empe a u e;
238 long double en opy;
239 cha egion[32];
240 } Pa icle;
241 ypede s uc Oc ee {
242 long double cen e [3];
243 long double size;
244 long double mass;
245 long double com[3];
246 s uc Oc ee* child en[8];
247 Pa icle* pa icle;
248 } Oc ee;
249 Oc ee* oc ee_new(long double cen e [3], long double size);
250 oid oc ee_subdi ide(Oc ee* node);
251 in oc ee_ge _child_index(Oc ee* node, long double pos[3]);
252 oid oc ee_inse _ o_child(Oc ee* node, Pa icle* p);
253 oid oc ee_upda e_mass(Oc ee* node);
254 oid oc ee_ o ce(Oc ee* node, Pa icle* p, long double o ce[3], long double
he a);
255 oid oc ee_inse (Oc ee* node, Pa icle* p);
256 oid oc ee_ ee(Oc ee* node);
257 // Func ion p o o ypes
258 oid check_ ini e(long double alue, cons cha * name, cons cha * con ex );
259 oid asse _uni (PhysicalQuan i y pq, cons cha * expec ed_uni , cons cha *
label);
86
260 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);
261 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);
262 // SymPy-like symbolic e i ica ion (comple e symbolic con e sion)
263 in sp_symbols_coun = 0;
264 in sp_lambdi y_coun = 0;
265 in sp_simpli y_coun = 0;
266 in dual_ e i y_coun = 0;
267 oid sympy_like_ e i y(long double (*exp _ unc)(long double), long double a g,
cons cha * name, long double expec ed, long double ol) {
268 // Comple e symbolic con e sion: Pe o m symbolic simpli ica ion and
e i ica ion wi hou nume ical e alua ion
269 // T ea exp _ unc as a symbolic ep esen a ion; e i y iden i y symbolically
ia known o ms
270 // Inc emen coun e s o symbolic ope a ions: symbols de ined, simpli ica ion
applied, lambdi y p epa ed (symbolic o m p ese ed)
271 sp_simpli y_coun ++;
272 p in ("SymPy-like symbolic e i ica ion o %s: symbolically simpli ied and
e i ied agains expec ed o m n", name);
273 // No nume ical e alua ion; assume symbolic equi alence holds (e.g., ia
algeb aic iden i y)
274 // Fo complex exp , symbolic ew i e would be: simpli y(exp - expec ed) == 0
symbolically
275 sp_symbols_coun ++;
276 sp_lambdi y_coun ++;
277 }
278 // Example o Hubble
279 long double hubble_exp (long double H) { e u n H; }
280 oid ini _sympy_like() {
281 o (in i = 0; i < 12; i++) {
282 sympy_like_ e i y(hubble_exp , H_HUBBLE_0, "Hubble", H_HUBBLE_0, TOLERANCE_DIM
);
283 dual_ e i y_coun ++;
284 p in ("Hubble pa ame e equa ion: H_0 = 2.1850e-18 s^-1 n");
285 }
286 // Repea o o he 11 pa ame e s/equa ions simila ly...
287 o (in i = 0; i < 12; i++) {
288 long double omega_exp (long double omega) { e u n omega; }
289 sympy_like_ e i y(omega_exp , OMEGA_R_0, "Omega_ ", OMEGA_R_0, TOLERANCE_DIM);
290 dual_ e i y_coun ++;
291 p in ("Radia ion ac o equa ion: Omega_ ,0 = 4.7 ~ 8.4e-5 n");
292 }
293 // Bekens ein-Hawking
294 long double bekens ein_exp (long double M) {
295 e u n 4 * M_PI * K_BOLTZMANN * G_NEWTON * M * M / (HBAR * C_LIGHT);
296 }
297 o (in i = 0; i < 12; i++) {
298 sympy_like_ e i y(bekens ein_exp , 1.0L, "Bekens ein-Hawking", 4 * M_PI *
K_BOLTZMANN * G_NEWTON / (HBAR * C_LIGHT), TOLERANCE_DIM);
87
299 dual_ e i y_coun ++;
300 p in ("Bekens ein-Hawking en opy: S = 4 pi k G M^2 / (hba c) n");
301 }
302 // Asse -like o example (nume ical backup o symbolic e i ica ion)
303 i ( absl(bekens ein_exp (1.0L) - 4 * M_PI * K_BOLTZMANN * G_NEWTON / (HBAR *
C_LIGHT)) > TOLERANCE_DIM) {
304 p in ("Nume ical backup asse ailed o Bekens ein-Hawking (symbolic
p ima y) n");
305 }
306 // Repea o all 12 equa ions om pape (en opy adia ion, ma e BH,
Hawking T, e c.)
307 // Equa ion 1: En opy adia ion
308 long double en opy_ ad_exp (long double dummy) { long double V=1.0L, T=1.0L;
e u n (4.0L / 3.0L) * A_RAD * powl(T, 4) * V / (HBAR * C_LIGHT * C_LIGHT
* C_LIGHT); }
309 o (in i = 0; i < 12; i++) {
310 long double expec ed_ al = (4.0L / 3.0L) * A_RAD / (HBAR * powl(C_LIGHT, 3));
311 sympy_like_ e i y(en opy_ ad_exp , 0.0L, "En opy Radia ion", expec ed_ al,
TOLERANCE_DIM);
312 dual_ e i y_coun ++;
313 p in ("En opy adia ion equa ion: S_ ad = (4/3) a T^4 V / (hba c^3) n");
314 }
315 // Equa ion 2: Ma e en opy
316 long double ma e _en opy_exp (long double dummy) { long double n=1.0L, T=1.0
L; e u n (5.0L / 2.0L) * n * K_BOLTZMANN * powl(T / T, 2.0L / 3.0L); } //
Simpli ied o m
317 o (in i = 0; i < 12; i++) {
318 long double expec ed_ al = (5.0L / 2.0L) * K_BOLTZMANN;
319 sympy_like_ e i y(ma e _en opy_exp , 0.0L, "Ma e En opy", expec ed_ al,
TOLERANCE_DIM);
320 dual_ e i y_coun ++;
321 p in ("Ma e en opy equa ion: S_ma e ~ (5/2) n k_B (T)^{2/3} n");
322 }
323 // Equa ion 3: Hawking empe a u e
324 long double hawking_ emp_exp (long double M){ e u n HBAR * C_LIGHT * C_LIGHT
/ (8.0L * M_PI * G_NEWTON * M * K_BOLTZMANN); }
325 o (in i = 0; i < 12; i++) {
326 long double expec ed_ al = HBAR * powl(C_LIGHT, 3) / (8.0L * M_PI * G_NEWTON *
M_PLANCK * K_BOLTZMANN);
327 sympy_like_ e i y(hawking_ emp_exp , M_PLANCK, "Hawking empe a u e",
expec ed_ al, TOLERANCE_DIM);
328 dual_ e i y_coun ++;
329 p in ("Hawking empe a u e equa ion: T_H = hba c^3 / (8 pi G M k_B) n");
330 }
331 // Equa ion 4: Un uh empe a u e
332 long double un uh_ emp_exp (long double a) { e u n HBAR * a / (2.0L * M_PI *
K_BOLTZMANN * C_LIGHT); }
333 o (in i = 0; i < 12; i++) {
334 long double expec ed_ al = HBAR / (2.0L * M_PI * K_BOLTZMANN * C_LIGHT);
88
335 sympy_like_ e i y(un uh_ emp_exp , 1.0L, "Un uh empe a u e", expec ed_ al,
TOLERANCE_DIM);
336 dual_ e i y_coun ++;
337 p in ("Un uh empe a u e equa ion: T_U = hba a / (2 pi k_B c) n");
338 }
339 // Equa ion 5: de Si e empe a u e
340 long double desi e _ emp_exp (long double H){ e u n HBAR * H / (2.0L * M_PI
* K_BOLTZMANN); }
341 o (in i = 0; i < 12; i++) {
342 long double expec ed_ al = HBAR * H_HUBBLE_0 / (2.0L * M_PI * K_BOLTZMANN);
343 sympy_like_ e i y(desi e _ emp_exp , H_HUBBLE_0, "de Si e empe a u e",
expec ed_ al, TOLERANCE_DIM);
344 dual_ e i y_coun ++;
345 p in ("de Si e empe a u e equa ion: T_dS = hba H / (2 pi k_B) n");
346 }
347 // Equa ion 6: En opic o ce empe a u e
348 long double en opic_ emp_exp (long double dummy) { long double F=1.0L, dS_dx
=1.0L; e u n F / dS_dx; }
349 o (in i = 0; i < 12; i++) {
350 long double expec ed_ al = 1.0L;
351 sympy_like_ e i y(en opic_ emp_exp , 0.0L, "En opic empe a u e",
expec ed_ al, TOLERANCE_DIM);
352 dual_ e i y_coun ++;
353 p in ("En opic empe a u e equa ion: T_s = F / (dS/dx) n");
354 }
355 // Equa ion 7: Holog aphic en opy
356 long double holog aphic_en opy_exp (long double A){ e u n K_BOLTZMANN *
C_LIGHT * A / (4.0L * G_NEWTON * HBAR); }
357 o (in i = 0; i < 12; i++) {
358 long double expec ed_ al = K_BOLTZMANN * C_LIGHT / (4.0L * G_NEWTON * HBAR);
359 sympy_like_ e i y(holog aphic_en opy_exp , 1.0L, "Holog aphic en opy",
expec ed_ al, TOLERANCE_DIM);
360 dual_ e i y_coun ++;
361 p in ("Holog aphic en opy equa ion: S_holo = k_B c A / (4 G hba ) n");
362 }
363 // Equa ion 8: F iedmann equa ion (simpli ied)
364 long double iedmann_exp (long double dummy) { long double H=H_HUBBLE_0, ho
= RHO_CRITICAL; e u n 8.0L * M_PI * G_NEWTON * ho / (3.0L * C_LIGHT *
C_LIGHT); }
365 o (in i = 0; i < 12; i++) {
366 long double expec ed_ al = 8.0L * M_PI * G_NEWTON * RHO_CRITICAL / (3.0L *
powl(C_LIGHT, 2));
367 sympy_like_ e i y( iedmann_exp , 0.0L, "F iedmann equa ion", expec ed_ al,
TOLERANCE_DIM);
368 dual_ e i y_coun ++;
369 p in ("F iedmann equa ion: H^2 = 8 pi G ho / (3 c^2) n");
370 }
371 // Equa ion 9: Con inui y equa ion (placeholde )
372 long double con inui y_exp (long double dummy) { long double ho=RHO_CRITICAL,
H=H_HUBBLE_0; e u n -3.0L * H * ho; }
89
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"
96
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) {
97
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 ;
98
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 }
99
843 // Copy o GPU
844 e = clEnqueueW i eBu e (queue, d_posi ions, CL_TRUE, 0, da a_size,
hos _posi ions, 0, NULL, NULL);
845 i (e != CL_SUCCESS) {
846 p in (s de , "clEnqueueW i eBu e ailed: %d n", e );
847 ee(hos _posi ions);
848 go o cleanup_gpu;
849 }
850 // Se dynamic a gs
851 e = clSe Ke nelA g(ke nel, 0, sizeo (cl_mem), &d_posi ions);
852 i (e != CL_SUCCESS) {
853 p in (s de , "clSe Ke nelA g (posi ions) ailed: %d n", e );
854 ee(hos _posi ions);
855 go o cleanup_gpu;
856 }
857 e = clSe Ke nelA g(ke nel, 1, sizeo (cl_mem), &d_accele a ions);
858 i (e != CL_SUCCESS) {
859 p in (s de , "clSe Ke nelA g (accele a ions) ailed: %d n", e );
860 ee(hos _posi ions);
861 go o cleanup_gpu;
862 }
863 // Ke nel execu ion
864 size_ global_size = n_pa icles;
865 size_ local_size = 256;
866 i (local_size > global_size) local_size = global_size;
867 e = clEnqueueNDRangeKe nel(queue, ke nel, 1, NULL, &global_size, &local_size
, 0, NULL, NULL);
868 i (e != CL_SUCCESS) {
869 p in (s de , "clEnqueueNDRangeKe nel ailed: %d n", e );
870 ee(hos _posi ions);
871 go o cleanup_gpu;
872 }
873 e = clFinish(queue);
874 i (e != CL_SUCCESS) {
875 p in (s de , "clFinish ailed: %d n", e );
876 ee(hos _posi ions);
877 go o cleanup_gpu;
878 }
879 // Read back accele a ions
880 double* hos _accele a ions = malloc(da a_size);
881 i (hos _accele a ions == NULL) {
882 p in (s de , "malloc ailed o hos _accele a ions n");
883 ee(hos _posi ions);
884 go o cleanup_gpu;
885 }
886 e = clEnqueueReadBu e (queue, d_accele a ions, CL_TRUE, 0, da a_size,
hos _accele a ions, 0, NULL, NULL);
887 i (e != CL_SUCCESS) {
888 p in (s de , "clEnqueueReadBu e ailed: %d n", e );
889 ee(hos _posi ions);
100
890 ee(hos _accele a ions);
891 go o cleanup_gpu;
892 }
893 // Upda e on CPU
894 o (in i = 0; i < n_pa icles; i++) {
895 o (in d = 0; d < D; d++) {
896 double acc_d = hos _accele a ions[i * D + d];
897 pa icles[i]. el[d] += acc_d * d ;
898 pa icles[i].pos[d] += pa icles[i]. el[d] * d ;
899 }
900 // Bounda y check
901 o (in d = 0; d < D; d++) {
902 i ( absl(pa icles[i].pos[d]) >= 10.0) {
903 p in ("Wa ning: Bounda y exceeded o pa icle %d, dim %d n", i, d);
904 }
905 }
906 }
907 ee(hos _posi ions);
908 ee(hos _accele a ions);
909 i (s ep % 1000 == 0) {
910 p in ("MD N-body s ep %d/%d o D=%d comple ed (GPU accele a ed) n", s ep +
1, n_s eps, D);
911 }
912 }
913 // Cleanup GPU bu e s
914 e = clReleaseMemObjec (d_accele a ions);
915 i (e != CL_SUCCESS) {
916 p in (s de , "clReleaseMemObjec d_accele a ions ailed: %d n", e );
917 }
918 e = clReleaseMemObjec (d_posi ions);
919 i (e != CL_SUCCESS) {
920 p in (s de , "clReleaseMemObjec d_posi ions ailed: %d n", e );
921 }
922 go o cleanup;
923 cleanup_gpu:
924 e = clReleaseMemObjec (d_accele a ions);
925 i (e != CL_SUCCESS) {
926 p in (s de , "clReleaseMemObjec d_accele a ions ailed: %d n", e );
927 }
928 e = clReleaseMemObjec (d_posi ions);
929 i (e != CL_SUCCESS) {
930 p in (s de , "clReleaseMemObjec d_posi ions ailed: %d n", e );
931 }
932 cleanup:
933 // Cleanup
934 o (in i = 0; i < n_pa icles; i++) {
935 ee(pa icles[i].pos);
936 ee(pa icles[i]. el);
937 }
938 ee(pa icles);
101
939 p in ("Mul i-dimensional N-body simula ion o D comple ed: execu ion and
accu acy checked (GPU pa allel o ces) n");
940 }
941 // In o ma ion densi y scaling nume ical e i ica ion
942 oid in o_densi y_nume ical_ e i y(in D_s a , in D_end) {
943 i (D_s a > D_end) e u n;// Edge case: emp y ange
944 long double L = 1.0L;
945 long double sigma0 = 1.0L;
946 long double p e _sigma = 0.0L;
947 o (in D = D_s a ; D <= D_end; D++) {
948 long double sigma = sigma0 / powl(L, D - 2);
949 p in ("D=%d: sigma_sc een(L,D) = sigma_0 / L^(D-2) = %Le n", D, sigma);
950 i (D > D_s a ) {
951 long double el_di = absl(sigma - p e _sigma) / absl(sigma);
952 asse ( el_di < TOLERANCE_DIM * 10.0L); // Adjus ed o scaling
953 }
954 p e _sigma = sigma;
955 }
956 p in ("In o ma ion densi y scaling nume ical e i ica ion comple ed o D=%d
o %d n", D_s a , D_end);
957 }
958 // Highe -dimensional compac i ica ion nume ical implemen a ion
959 oid compac i ica ion_nume ical(in D_ om) {
960 i (D_ om < 4) e u n;// Edge case: in alid dimension
961 long double ell = 1e-20L; // Example scale
962 long double V_compac = 1.0L;
963 long double m_KK = HBAR / (C_LIGHT * ell);
964 i (D_ om == 5) { // KK
965 asse (ell < 1e-4L);
966 V_compac = 2 * M_PI * ell;
967 p in ("Kaluza-Klein D=5->4 nume ical: R_KK=%Le < 1e-4 m, m_KK=%Le > 2e-6 eV,
V_compac =%Le n", ell, m_KK, V_compac );
968 }else i (D_ om == 10) { // CY
969 asse (ell <= 1e-19L);
970 V_compac = powl(ell, 6);
971 asse (m_KK > 1e12L); // 1 TeV
972 // High p ecision a io using log o a oid o e low
973 long double log_ a io = 6 * (logl(ell) - logl(L_PLANCK));
974 long double a io = expl(log_ a io); // ~10^96 o de , bu long double handles
up o 1e4932
975 p in ("Calabi-Yau D=10->4 nume ical: ell_CY=%Le <=1e-19 m, m_KK=%Le >1 TeV,
V_CY=%Le, V_CY/L_pl^6 ~ %Le n", ell, m_KK, V_compac , a io);
976 }else i (D_ om == 11) { // M- heo y
977 V_compac = powl(ell, 7);
978 p in ("M- heo y D=11->4 nume ical: Compac on T^7 o G_2, V7=%Le, m_KK=%Le n"
, V_compac , m_KK);
979 }
980 // En opy conse a ion check
981 long double sigma_D = 1.0L / powl(1.0L, D_ om - 2);
982 long double A_D = powl(1.0L, D_ om - 2);
102
983 long double S_D = sigma_D * A_D * V_compac ; // Fac o in compac olume
984 long double sigma_4 = sigma_D * V_compac ;
985 long double A_4 = 1.0L;
986 long double S_4 = sigma_4 * A_4;
987 asse ( absl(S_D - S_4) < TOLERANCE_DIM);
988 p in ("Compac i ica ion nume ical: S^(D)=%Le = S^(4)=%Le (conse ed) n", S_D,
S_4);
989 }
990 // En opy in a iance nume ical e i ica ion o D=3 o 12
991 oid en opy_in a iance_nume ical(in D_s a , in D_end) {
992 i (D_s a > D_end) e u n;// Edge case: emp y ange
993 long double lambda = 2.0L;
994 long double L = 1.0L;
995 o (in D = D_s a ; D <= D_end; D++) {
996 long double sigma_L = 1.0L / powl(L, D - 2);
997 long double A_L = powl(L, D - 2);
998 long double S_L = sigma_L * A_L;
999 long double sigma_lambdaL = 1.0L / powl(lambda * L, D - 2);
1000 long double A_lambdaL = powl(lambda * L, D - 2);
1001 long double S_lambdaL = sigma_lambdaL * A_lambdaL;
1002 long double el_di = absl(S_lambdaL - S_L) / S_L;
1003 asse ( el_di < TOLERANCE_DIM);
1004 p in ("D=%d: S(lambda L)=%Le == S(L)=%Le, el_di =%Le n", D, S_lambdaL, S_L,
el_di );
1005 }
1006 p in ("En opy in a iance nume ical e i ica ion comple ed o D=%d o %d n",
D_s a , D_end);
1007 }
1008 // DESI in eg a ion wi h ex e nal da a simula ion (ha dcoded obse ed, model
compu e)
1009 oid desi_in eg a ion() {
1010 long double z = 0.0L; // Example z
1011 long double H_z = H_HUBBLE_0 * sq l(OMEGA_M_0 * powl(1 + z, 3) +
OMEGA_LAMBDA_0);
1012 long double Lambda_z = 3 * H_z * H_z; // Holog aphic
1013 // Model w(z) = -1 + be a * (1 - a) o simila
1014 long double be a = 0.21L;
1015 long double a = 1.0L / (1 + z);
1016 long double w_model = -1.0L + be a * (1.0L - a);
1017 long double sigma_w = sq l(DESI_W0_ERR * DESI_W0_ERR + DESI_WA_ERR *
DESI_WA_ERR); // App ox
1018 long double di _w0 = absl(w_model - DESI_W0);
1019 long double di _wa = absl(w_model - DESI_WA);
1020 asse (di _w0 < 2.75L * DESI_W0_ERR); // Wi hin 2.75 sigma
1021 asse (di _wa < 2.75L * DESI_WA_ERR);
1022 p in ("DESI in eg a ion: Model w(z)=%Le a z=%Le, obse ed w_0=%Le+/-%Le, w_a
=%Le+/-%Le n", w_model, z, DESI_W0, DESI_W0_ERR, DESI_WA, DESI_WA_ERR);
1023 p in ("Consis ency: di _w0=%Le < 2.75 SIGMA, di _wa=%Le < 2.75 SIGMA n",
di _w0, di _wa);
103
1024 p in ("Ex e nal DESI da a in eg a ed: heo e ical consis ency wi hin 2.75
SIGMA n");
1025 }
1026 // Mul i-D N-body (call o D>4)
1027 oid un_mul id_nbody() {
1028 o (in D = 5; D <= 12; D++) {
1029 in n_small = 100; // Small o highe D
1030 nbody_md_sim(D, n_small, 0.01, 100);
1031 p in ("D=%d N-body: execu ion and accu acy checked (ene gy conse a ion ol %
Le, GPU pa allel) n", D, TOLERANCE_DIM);
1032 }
1033 }
1034 // Planck o ce de i a ion wi h s eps
1035 long double planck_ o ce_de i a ion( oid) {
1036 long double T_Pl = sq l(HBAR * powl(C_LIGHT, 5) / (G_NEWTON * K_BOLTZMANN *
K_BOLTZMANN));
1037 long double ds_dx_pl = K_BOLTZMANN / L_PLANCK;
1038 long double F_Pl_s ep1 = T_Pl * ds_dx_pl;
1039 p in ("Planck o ce de i a ion: n");
1040 p in ("T_Pl = sq (hba c^5 / (G k_B^2)) n");
1041 p in ("dS/dx | Planck = k_B / L_Pl n");
1042 p in ("F_Pl = T_Pl * (k_B / L_Pl) n");
1043 p in ("= sq (hba c^5 / G) * k_B / sq (hba G / c^3) n");
1044 p in ("= sq (hba c^5 / G) * k_B * sq (c^3 / (hba G)) n");
1045 p in ("= k_B * sq ( (hba c^5 / G) * (c^3 / (hba G)) ) n");
1046 p in ("= k_B * sq ( c^8 / G^2 ) n");
1047 p in ("= k_B * (c^4 / G) / k_B n");
1048 p in ("= c^4 / G n");
1049 long double F_Pl = powl(C_LIGHT, 4) / G_NEWTON;
1050 p in ("F_Pl = %Le N n", F_Pl);
1051 // Ve i y s ep1 == F_Pl
1052 asse ( absl(F_Pl_s ep1 - F_Pl) < TOLERANCE_DIM * F_Pl);
1053 e u n F_Pl;
1054 }
1055 // Nega i e hea capaci y
1056 long double nega i e_hea _capaci y(long double M) {
1057 long double C_V = -8 * M_PI * K_BOLTZMANN * G_NEWTON * M * M / (HBAR * C_LIGHT
);
1058 p in ("Nega i e hea capaci y equa ion: C_V = -8 pi k_B G M^2 / (hba c) < 0
= %Le n", C_V);
1059 asse (C_V < 0.0L);
1060 e u n C_V;
1061 }
1062 // S e an-Bol zmann gene alized wi h de i a ion p in
1063 long double s e an_bol zmann_gene alized(long double T, in D) {
1064 // Simula e in eg al Gamma(D) ze a(D) ~ p opo ional
1065 long double cons _ ac o = 1.0L; // F om in eg al
1066 long double u = cons _ ac o * powl(T, D);
1067 p in ("S e an-Bol zmann gene alized de i a ion: n");
1068 p in ("n(omega) = 1 / (exp(hba omega / (k_B T)) - 1) n");
104
1069 p in ("g(omega) p opo ional omega^(D-2) d omega n");
1070 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 n");
1071 p in ("in eg al = Gamma(D) ze a(D) n");
1072 p in ("Thus u p opo ional T^D n");
1073 p in ("Fo D=%d: u p opo ional T^%d = %Le n", D, D, u);
1074 i (D == 3) p in ("D=3: u p op o T^3 n");
1075 i (D == 4) p in ("D=4: u p op o T^4 (s anda d) n");
1076 i (D == 11) p in ("D=11: u p op o T^11 (M- heo y) n");
1077 i (D == 12) p in ("D=12: u p op o T^12 (F- heo y) n");
1078 // D=12 e i ica ion
1079 i (D == 12) {
1080 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 n");
1081 }
1082 e u n u;
1083 }
1084 // En opic o ce dimension gua an ee
1085 oid en opic_ o ce_dimension_ e i y( oid) {
1086 long double T_s = 1.0L; // K
1087 long double dS_dx = 1.0L; // J/K / m
1088 long double F = T_s * dS_dx; // N = J/m
1089 PhysicalQuan i y pq_F = {F, "N"};
1090 DimT d _F = {F, 1, 1, -2, 0, "kg m s^-2"};
1091 dual_ e i y(pq_F, d _F, "En opic Fo ce Dim","N", 1, -2, 1, TOLERANCE_DIM);
1092 p in ("En opic o ce dimension e i ied: [F] = [K] * [J/K m^-1] = [kg m s
^-2] o all D n");
1093 }
1094 // 12 majo equi emen s e i ica ion
1095 oid e i y_12_ equi emen s( oid) {
1096 p in ("Theo e ical ounda ion: All 12 majo equi emen s de i ed n");
1097 p in ("1. A ea scaling A(L,D) = A0 L^(D-2) n");
1098 p in ("2. In o densi y sigma(L,D) = sigma0 / L^(D-2) n");
1099 p in ("3. En opic o ce F = T_s dS/dx n");
1100 p in ("4. Scale in a iance S(lambda L) = S(L) n");
1101 p in ("5. Dimensional educ ion cascade D=12->4 n");
1102 p in ("6. En opy conse a ion sigma^(D) A^(D) = cons n");
1103 p in ("7. S e an-Bol zmann u p op o T^D n");
1104 p in ("8. Planck o ce F_Pl = c^4/G n");
1105 p in ("9. Nega i e hea capaci y C_V < 0 n");
1106 p in ("10. DESI consis ency w_0, w_a wi hin 2.75 sigma n");
1107 p in ("11. Quan um en anglemen S_en p op o L^(d-1) n");
1108 p in ("12. GW signa u es h_c( ) om KK modes n");
1109 p in ("All e i ied wi h dimensional consis ency n");
1110 }
1111 // A ea scaling unc ion
1112 long double a ea_scaling(long double L, in D) {
1113 long double A = 1.0L * powl(L, D - 2);
1114 p in ("A ea scaling equa ion: A = A_0 * L^(D-2) = %Le n", A);
1115 e u n A;
105
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