Six Unified Physics Problems as Consistency Constraints of the Unified Matrix--QCA Universe \large Common Solution Space for Black Holes, Cosmological Constant, Neutrinos, ETH, Strong CP, and Gravitational Wave Dispersion

Six Unied Physics P oblems as Consis ency Cons ain s o he
Unied Ma ixQCA Uni e se
Common Solu ion Space o Black Holes, Cosmological Cons an , Neu inos, ETH, S ong
CP, and G a i a ional Wa e Dispe sion
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
Wi hin he amewo k o unied ime scale, bounda y ime geome y, THE-MATRIX ma ix
uni e se, and quan um cellula au oma on (QCA) uni e se, he six p oblemsblack hole en opy
and in o ma ion pa adox, cosmological cons an and da k ene gy, neu ino mass and a o
mixing, quan um chaos and eigens a e he maliza ion hypo hesis (ETH), s ong CP p oblem
and axion, g a i a ional wa e Lo en z iola ion and dispe sionha e been sepa a ely embedded
in o local s uc u es. Howe e , om he pe spec i e o a unied uni e sal objec , hese six
p oblems should no be iewed as six independen sub opics, bu a he ew i en as six se s o
consis ency cons ain s on he same uni e sal mo he objec .
This pape in oduces a ni e-dimensional s uc u al pa ame e amily in he con ex o
unied uni e sal objec
U⋆
phys
, including QCA cell la ice spacing and ime s ep
(ℓcell,∆ )
, local
dimension and decomposi ion o cell Hilbe space, p ojec ion o unied ime scale densi y
κ(ω)
in die en equency bands and channel sec o s, componen s o ela i e cohomology class
[K]∈H2(Y, ∂Y ;Z2)
, and design pa ame e s o axioma ically chao ic QCA. The six unied
physics p oblems a e uni o mly o mula ed as six cons ain s on his pa ame e se : black hole
en opy p o ides ela ion be ween ho izon cell en opy densi y and la ice spacing; cosmological
cons an p oblem p o ides windowed spec al sum ule o unied ime scale densi y; neu ino
mass and a o mixing p o ide geome ized cons ain s on a o QCA seesaw mass ma ix and
PMNS holonomy; ETH equi es QCA o be axioma ically chao ic QCA in each ni e causal
diamond; s ong CP p oblem cons ains opological wis o sca e ing de e minan line bundle
squa e oo ia
[K]=0
; g a i a ional wa e dispe sion p o ides obse a ional uppe bound on
ℓcell
and dispe sion coecien s
β2n
, o bidding odd-o de dispe sion e ms.
Based on hese cons ain s, his pape p esen s a s uc u al heo em: unde na u al scale
hie a chy and locali y assump ions, a pa ame e poin amily
(ℓcell,dim Hcell, κ, [K], β2n, . . . )
exis s simul aneously sa is ying all six cons ain s, hus he six p oblems possess non-emp y
common solu ion space in he unied uni e sal amewo k. Appendices p o ide p oo ou lines
o key heo ems and p o o ype solu ion cons uc ion example.
Keywo ds
: Unied ime scale; ma ix uni e se; quan um cellula au oma on; black hole en-
opy; cosmological cons an p oblem; neu ino mass and PMNS ma ix; eigens a e he maliza ion
hypo hesis (ETH); s ong CP p oblem and axion; g a i a ional wa e dispe sion; spec al unc ion
sum ule
1 In oduc ion & His o ical Con ex
Black hole en opy, cosmological cons an , neu ino mass, quan um chaos, s ong CP p oblem, and
g a i a ional wa e dispe sion cons i u e he mos p ominen se o " esidual p oblems" in con em-
1
po a y high-ene gy physics and cosmology. They espec i ely poin owa d six di ec ionsg a i y,
quan um eld heo y, a o physics, non-equilib ium s a is ics, opology, and g a i a ional wa e
obse a ionsye hese p oblems a e highly in e wined: black hole en opy and in o ma ion pa a-
dox in ol e mic os a e coun ing and uni a i y in quan um g a i y; cosmological cons an p oblem
connec s acuum ene gy densi y na u alness wi h g a i a ional edshi and la ge-scale s uc u e;
neu ino mass and a o mixing equi e in oduc ion o seesaw mechanism and a o symme y be-
yond S anda d Model; ETH is co e mechanism o unde s anding he maliza ion in isola ed many-
body sys ems; s ong CP p oblem e eals ne balance be ween QCD opology and CP symme y;
g a i a ional wa e dispe sion cons ains possible modica ions o gene al ela i i y a p opaga ion
le el.
In black hole physics, ea ly wo ks by Bekens ein and Hawking showed ha he modynamic
en opy o s a ic black holes sa ises a ea law
SBH =A/(4G)
, consolida ed om Euclideanized
GibbonsHawking pa h in eg al, Wald en opy o mula, and la e mic os a e coun ing amewo ks
[13].
Cosmological cons an p oblem was sys ema ically cha ac e ized in Weinbe g's classic e iew:
obse a ional cosmological cons an
Λobs
is smalle han nai e acuum ene gy es ima e by abou
10120
o de s o magni ude, a disc epancy dicul o handle ia con en ional eld heo y eno -
maliza ion [4]. Subsequen ly appea ed se ies o wo ks based on spec al unc ions and sum ules,
ew i ing UV acuum ene gy con ibu ion as spec al in eg al achie ing cancella ion ia high-ene gy
spec al ha monic condi ion [5,6].
Neu ino oscilla ion expe imen s (Supe -Kamiokande, SNO, Daya Bay, T2K, NO A, e c.) indi-
ca ed neu inos ha e non-ze o mass, wi h a o eigens a es
(νe, νµ, ντ)
ela ed o mass eigens a es
(ν1, ν2, ν3)
ia PMNS ma ix [7,8]. PDG neu ino chap e and mul iple e iews summa ize h ee-
a o mixing pa ame e s uc u e, mass-squa ed die ences, and CP phase s a us [9].
In quan um s a is ics, ETH was p oposed o explain he maliza ion beha io o isola ed many-
body sys ems: o chao ic sys ems' o e whelming majo i y o ene gy eigens a es, local obse able
expec a ion alues in he modynamic limi consis en wi h he mal equilib ium ensemble esul s
[10]. Sys ema ic ETH e iews connec i wi h quan um chaos, andom ma ix heo y, and sel -
consis en he modynamic ela ions, p o iding abundan nume ical e idence om la ice models
[11,12].
S ong CP p oblem o igina es om QCD
θ
- e m: expe imen al cons ain ia neu on elec ic
dipole momen indica es physical
¯
θ
ex emely small, while S anda d Model seemingly allowed CP
iola ion equi es his angle na u ally
O(1)
. PecceiQuinn mechanism and axion eld conside ed
among mos powe ul solu ions, wi h e iews de ailing s ong CP p oblem s a us and a ious axion
physics implemen a ions [13,14].
Finally, g a i a ional wa e mul i-messenge obse a ions (especially GW170817 and GRB170817A
join de ec ion) p o ided ex emely s ingen cons ain s on g a i a ional wa e p opaga ion speed,
supp essing
| g/c −1|
o
10−15
o smalle [15]. Subsequen obse a ions and simula ion s udies
p o ided bounds on dispe sion pa ame e s in mo e gene al modied g a i y models, cons i u ing
s ong cons ain s on any disc e e o modied quan um g a i y scheme [16].
On he o he hand, disc e e uni e se ideas ep esen ed by causal se s, disc e e eld heo y, quan-
um cellula au oma a ha e de eloped ex ensi ely in ecen yea s, a emp ing o econs uc con-
inuous space ime and quan um eld heo y om p imi i e disc e e s uc u es [1719]. In p e ious
wo ks, an o e all amewo k has been cons uc ed: unied ime scale
κ(ω)
, bounda y ime geome y
(BTG), THE-MATRIX ma ix uni e se, and QCA uni e se
Uqca
, whe e unied ime scale de i ed
om alignmen o sca e ing hal -phase de i a i e, ela i e densi y o s a es, and Wigne Smi h
g oup delay ace; bounda y ime geome y unies modula ow, geome ic ime, and sca e ing
ime scale; ma ix uni e se and QCA uni e se equi alen o geome icQFT uni e se in con inuum
2
limi .
Wi hin his amewo k, he a o emen ioned six physics p oblems ha e been sepa a ely es a ed
as p ope ies o ollowing s uc u al laye s:
1. Black hole en opy and in o ma ion eco e y: en anglemen en opy a ea law and uni a y
Page cu e on ho izon band QCA;
2. Cosmological cons an and da k ene gy: windowed spec al in eg al and high-ene gy DOS
sum ule o unied ime scale densi y;
3. Neu ino mass and a o mixing: sca e ing holonomy and QCA seesaw s uc u e on a o 
bundles;
4. ETH and quan um chaos: local uni a y design and ypicali y o axioma ically chao ic QCA;
5. S ong CP and axion: ela i e cohomology class
[K]∈H2(Y, ∂Y ;Z2)
QCD componen
[KQCD]
and sca e ing de e minan line bundle squa e oo wis ;
6. G a i a ional wa e dispe sion: cons ain s on e en-o de
(kℓcell)2n
co ec ions in g a i y
QCA dispe sion ela ion and exclusion o odd-o de e ms.
Howe e , as long as hese p oblems a e ea ed sepa a ely, a global conclusion abou "whe he
he same uni e sal objec can simul aneously sa is y all cons ain s" is s ill lacking. In o he wo ds,
e en i each subp oblem has na u al o mula ion in unied amewo k, p o ing exis ence o a class o
unied uni e sal objec s making six ypes o phenomena mu ually compa ible on he same s uc u al
pa ame e se is equi ed.
This pape 's goal is o uni o mly ew i e abo e six p oblems as six se s o cons ain equa ions
on a ni e-dimensional pa ame e amily, p o ing unde na u al assump ions his cons ain sys em
possesses non-emp y solu ion space, he eby ele a ing six unied physics p oblems o consis ency
condi ions o unied ma ixQCA uni e se, a he han six isola ed puzzles.
2 Model & Assump ions
This sec ion  s b iey e iews unied uni e sal mo he objec
U⋆
phys
, hen ex ac s ni e-dimensional
pa ame e amily epea edly used in subsequen analysis, p o iding basic axioms and wo king as-
sump ions.
2.1 Unied Uni e sal Mo he Objec
U⋆
phys
Unied uni e sal objec deno ed as
U⋆
phys =Ue , Ugeo, Umeas, UQFT, Usca , Umod, Uen , Uobs, Uca , Ucomp, UBTG, Uma , Uqca, U op.
Componen meanings as ollows.
1. E en and geome y laye :
*
Ue
: e en se and causal pa ial o de , sa is ying global hype bolici y;
*
Ugeo = (M, g, ≺)
: ou -dimensional globally hype bolic Lo en zian mani old, compa ible wi h
Ue
pa ial o de .
2. Field heo y and sca e ing laye :
*
UQFT
: quan um eld heo y on cu ed space ime and eec i e ac ion
Se [g, A, ψ, ϕ]
;
*
Usca
: sca e ing pai
(H, H0)
, sca e ing ma ix
S(ω)
, spec al shi unc ion
ξ(ω)
, and unied
ime scale densi y
κ(ω) = 1
2π Q(ω), Q(ω) = −iS(ω)†∂ωS(ω).
3
3. Modula and en opy laye :
*
Umod
: Tomi aTakesaki modula ow on bounda y obse able algeb a;
*
Uen
: gene alized en opy unc ion amily and QNEC/QFC ype inequali ies.
4. Obse e and ca ego y laye :
*
Uobs
: collec ion o obse e wo ldlines, accessible causal domains, and upda e ules;
*
Uca
: s uc u e o ganizing geome icsca e ingobse a ion p ocess in o 2-ca ego y;
*
Ucomp
: abs ac cha ac e iza ion o uni e se as compu a ional p ocess.
5. Bounda y and ma ix laye :
*
UBTG
: bounda y obse able algeb a
A∂
, s a e
ω∂
, modula ow
σω
, and bounda y ime
geome y aligned wi h
κ(ω)
;
*
Uma
: channel Hilbe space di ec sum
Hchan
and equency-decomposed sca e ing ma ix
amily
S(ω)
, cons i u ing THE-MATRIX ma ix uni e se.
6. QCA and opology laye :
*
Uqca
: uni e sal QCA objec
(Λ,Hcell,Aqloc, α, ω0)
, whe e
Λ
is coun able connec ed g aph (e.g.,
hype cubic la ice),
Aqloc
quasilocal
C∗
algeb a,
α
disc e e amily o
∗
-au omo phisms wi h ni e
p opaga ion adius,
ω0
ini ial uni e sal s a e;
*
U op
: ela i e cohomology class
[K]∈H2(Y, ∂Y ;Z2)
on ex ended space imepa ame e space
Y=M×X◦
, cha ac e izing
Z2
wis o sca e ing de e minan line bundle squa e oo
L1/2
de
and
NullModula double co e consis ency.
P e ious wo k es ablished: in app op ia e uni e sal ca ego y, objec s sa is ying unied ime
scale, gene alized en opy mono onici y, and local quan um g a i y cons ain s can be embedded
in o some
U⋆
phys
, wi h his embedding possessing e minal objec p ope y in na u al 2-mo phism
sense.
2.2 Uni e sal QCA and Con inuum Limi Assump ion
Uni e sal QCA objec deno ed as
Uqca = (Λ,Hcell,Aqloc, α, ω0),
whe e:
*
Λ
deg ee-bounded coun able connec ed g aph (in his wo k ake
Λ≃Z3
hype cubic la ice);
* Each cell ca ies ni e-dimensional Hilbe space
Hcell
, o al Hilbe space inni e enso
p oduc ;
*
Aqloc
quasilocal
C∗
algeb a gene a ed by local ope a o s;
*
α:Z→Au (Aqloc)
ime e olu ion wi h ni e p opaga ion adius, au omo phism has uni a y
ealiza ion
U
.
We adop ollowing con inuum limi assump ion.
Assump ion 1
(2.1: QCAGeome y Con inuum Limi )
.
Scale pa ame e s
ℓcell >0
,
∆ > 0
exis ,
along wi h coa se-g aining scheme, such ha unde app op ia e eno maliza ion limi ,
(Λ,Hcell, α)
long-wa eleng h dynamics equi alen o eec i e QFT
UQFT
on
(M, g)
, wi h unied ime scale
densi y
κ(ω)
econs uc ible om QCA band s uc u e spec al da a.
Resea ch by T ezzini e al. on QCA coa se-g aining and mul iple causal disc e e eld heo y
schemes indica e cons uc ing QCA wi h easonable con inuum limi unde ni e p opaga ion adius
and locali y condi ions is easible [1719].
4
2.3 S uc u al Pa ame e Family
To uni o mly cha ac e ize six physics p oblems, in oduce ollowing s uc u al pa ame e amily:
p=ℓcell,∆ , dcell,H
decomp
cell , κ(ω)
sec o
,[K], β2n,
ETH da a
,
a o da a
.
Specically including:
1.
Disc e e geome y pa ame e s
ℓcell
as QCA la ice spacing,
∆
as ime s ep.
2.
Local Hilbe dimension and decomposi ion
Hcell ≃ Hg a ⊗ Hgauge ⊗ Hma e ⊗ Haux, dcell = dim Hcell.
Fo a o and neu ino sec o , ake
H(ν)
cell ≃C3⊗ Hspin ⊗ Haux
.
3.
Unied ime scale densi y sec o s uc u e
κ(ω) = X
a
κa(ω), a ∈g a ,QCD, la o , ad, . . .,
along wi h co esponding DOS die ence
∆ρ(E)
.
4.
Topological class and CP pa ame e
[K]∈H2(Y, ∂Y ;Z2),[K]=[Kg a ]+[KEW]+[KQCD] + · · · ,
and QCD sec o eec i e angle
¯
θe
.
5.
Axioma ically chao ic QCA pa ame e s
Including local ga e se , p opaga ion adius
R
, app oxima e uni a y design o de
, local ene gy
shell en opy densi y
s(ε)
, spec al non-degene acy, e c.
6.
Dispe sion pa ame e s
In g a i yQCA dispe sion ela ion
ω2=c2k2h1 + X
n≥1
β2n(kℓcell)2ni
coecien s
β2n
, odd-o de e ms excluded by unied amewo k.
2.4 Wo king Assump ions and Technical Condi ions
Subsequen heo ems equi e se e al echnical assump ions:
1.
Con ollabili y o hea ke nel and spec al shi
Sca e ing pai
(H, H0)
sa ises ace-class condi ion and Bi manK en o mula, enabling Taube-
ian co espondence be ween hea ke nel die ence
∆K(s) = (e−sH −e−sH0)
and spec al shi unc ion
ξ(ω)
.
2.
QCA band s uc u e egula i y
Band s uc u e in UV egion app oximable by ni e numbe o smoo h band unc ions
εj(k)
,
DOS die ence
∆ρj(k)
a ies smoo hly wi h
k
, suppo ing spec al unc ion sum ule ew i ing [5,6].
3.
Seesaw implemen a ion in a o QCA
Local QCA upda e subblock exis s,
Uloc
x= exph−i∆ 0MD(x)
M†
D(x)MR(x)i,
5

yielding seesaw mass ma ix in con inuum limi
Mν=−MT
DM−1
RMD
. This s uc u e is s anda d cons uc ion in mul iple seesaw models and
a o symme y implemen a ions [79].
4.
Axioma ically chao ic QCA hypo hesis
In each ni e causal diamond, QCA es ic ion
UΩ
app oximable by ni e-dep h local andom
ci cui , local ga e se gene a ing app oxima e Haa dis ibu ion, he eby implemen ing ETH on
local obse ables [1012].
5.
Axion and opological line bundle hypo hesis
QCD
θ
- e m and Yukawa phase uni o mly encoded in sca e ing de e minan line bundle squa e
oo
L1/2
de
,
Z2
wis con olled by ela i e cohomology class
[K]
QCD componen
[KQCD]
, consis en
wi h exis ing opological unde s anding o s ong CP p oblem and axion eec i e heo y [13,14].
6.
Applicabili y o g a i a ional wa e dispe sion cons ain s
Adop p opaga ion speed cons ain s om GW170817/GRB170817A and subsequen e en s,
con e ing
| g/c −1|≲10−15
in o uppe bound on
β2ℓ2
cell
[15,16].
Unde hese assump ions, six physics p oblems can be uni o mly w i en as cons ain s on pa-
ame e amily
p
, p o ing hei common solu ion space non-emp y.
3 Main Resul s (Theo ems and Alignmen s)
This sec ion p esen s unied o mula ion o six physics p oblems on pa ame e amily
p
, o ganizing
hem as heo em se . Fo b e i y, all heo ems unde s ood unde Sec ion 2 assump ions and echnical
condi ions.
3.1 Black Hole En opy and Ho izon Cell Cons ain
Theo em 2
(3.1: Black Hole En opy and G a i yQCA La ice Spacing)
.
Assume QCA uni e se
con ains ho izon band subla ice
ΓH⊂Λ
, whose embedding app oxima es geome ic ho izon sec ion
ΣH
, sa is ying
NH:= |ΓH|=A(ΣH)
ℓ2
cell
+O(A0),
ho izon Hilbe space
HH≃ H⊗NH
g a
, ypical equilib ium s a es highly en angled wi hin ene gy shell,
hen c oss-ho izon en anglemen en opy
Sen (ΣH) = ηg a
A(ΣH)
ℓ2
cell
+O(A0), ηg a = log de ≤log dg a .
I equi ing gene alized en opy sa is y Bekens einHawking a ea law
SBH =A/(4G) + O(A0)
,
hen mus and need only sa is y
ηg a
ℓ2
cell
=1
4G,
i.e.
ℓ2
cell = 4Glog de .
In o he wo ds, black hole en opy in QCA uni e se equi alen o cons ain cu e on
(ℓcell, dg a )
,
na u ally xing la ice spacing a Planck scale o de .
6
3.2 Cosmological Cons an and Windowed Spec al Sum Rule
Theo em 3
(3.2: Cosmological Cons an Unied Time Scale Sum Rule)
.
Assume unied ime scale
densi y
κ(ω)
sa ises phasespec al-shi g oup-delay chain
κ(ω) = φ′(ω)/π =−ξ′(ω) = (2π)−1 Q(ω)
, app op ia e loga i hmic window ke nel
W(ln(ω/µ))
exis s, hen cosmological cons an eec i e inc emen w i able as
Λe (µ)−Λe (µ0) = Zµ
µ0
Ξ(ω) d ln ω,
whe e
Ξ(ω)
windowed unc ion o
κ(ω)
. I QCA band s uc u e sa ises high-ene gy spec al sum
ule in UV egion
ZEUV
0
E2∆ρ(E) dE= 0,∆ρ(E)=∆ρκ(ω),
band s uc u e
,
hen windowed high-ene gy acuum ene gy con ibu ions mu ually cancel in
Λe
, lea ing only
ni e esidual de e mined by IR scale
EIR
, magni ude
Λe ∼E4
IREIR
EUV γ, γ > 0.
The e o e, cosmological cons an p oblem in unied amewo k equi alen o high-ene gy DOS
die ence sa is ying abo e sum ule, p o iding second cons ain on
κ(ω)
beha io in UV egion.
3.3 Neu ino Mass and Fla o QCA Seesaw Cons ain
Theo em 4
(3.3: PMNS Holonomy and Seesaw Mass Ma ix QCA Implemen a ion)
.
Assume
lep onic sec o cell Hilbe space decomposes as
H(ν)
cell ≃C3⊗ Hspin ⊗ Haux,
local QCA upda e includes seesaw block
Uloc
x
in a o subspace, yielding Majo ana mass ma ix
in con inuum limi
Mν=−MT
DM−1
RMD
. Dene a o connec ion in equency space
A la o (ω) = U†
PMNS(ω)∂ωUPMNS(ω),
hen holonomy along unied ime scale pa h
γcc
Uγcc =Pexp−Zγcc
A la o (ω) dω∼UPMNS.
Unde abo e condi ions, s anda d h ee-a o neu ino oscilla ion da a and seesaw mass spec-
um ealizabili y equi alen o a o QCA seesaw module and connec ion
A la o
sa is ying expe -
imen ally de e mined PMNS ex u e and mass-squa ed die ence cons ain s. This p o ides hi d
cons ain on
H(ν)
cell
,
MD, MR
, and
κ(ω)
in a o window.
7
3.4 ETH and Axioma ically Chao ic QCA Cons ain
Theo em 5
(3.4: Local ETH o Axioma ically Chao ic QCA)
.
Assume on a bi a y ni e egion
Ω⊂Λ
, QCA es ic ion
UΩ
app oximable by ni e-dep h local andom ci cui , local ga e se gene -
a ing app oxima e
-o de uni a y design a e se e al laye s, sys em ha ing only ene gy and ni e
global quan um numbe conse a ion, hen o a bi a y local ope a o
OX
(
X⊂Ω
), almos all
quasi-ene gy eigens a es
|ψn⟩
sa is y
⟨ψn|OX|ψn⟩=OX(εn) + Oe−c|Ω|,
o-diagonal elemen squa ed a e age simila ly exponen ially decays wi h olume. He e
OX(ε)
mic ocanonical a e age a ene gy densi y
ε
. This p ope y cons i u es ETH, h ough ela ion be ween
unied ime scale
κ(ω)
and QCA ene gy spec um, making mac oscopic he mal ime a ow ypical
beha io o QCA uni e se, a he han addi ional pos ula e.
The e o e, ETH es ablishmen in unied uni e se equi alen o QCA sa is ying axioma ically
chao ic condi ion in each ni e causal diamond, ou h cons ain on "ETH da a" in pa ame e
amily
p
.
3.5 S ong CP P oblem and Topological Class
[K]
Cons ain
Theo em 6
(3.5: S ong CP and Rela i e Cohomology Class T i iali y)
.
Assume QCD sec o
θ
- e m, Yukawa phase, and o he CP phases uni o mly encoded in sca e ing de e minan line bun-
dle squa e oo
L1/2
de
,
Z2
wis ep esen ed by ela i e cohomology class
[K]∈H2(Y, ∂Y ;Z2)
QCD
componen
[KQCD]
. I equi ing:
1. No hal -ci cle phase anomaly in NullModula double co e ;
2. Equi alence be ween bounda y gene alized en opy ex emum and Eins ein equa ions holds;
3. Physical
¯
θe
supp essed o cu en expe imen al cons ain ange,
hen opological sec o mus exis making
[K] = 0,
pa icula ly
[KQCD] = 0,
hen in some global choice can abso b
¯
θe
in o squa e oo gauge choice; PecceiQuinn axion
eld in his sec o in e p e able as
U(1)
be coo dina e on
L1/2
de
, eec i e po en ial minimum au o-
ma ically implemen ing
¯
θe = 0
. Con e sely i
[KQCD]= 0
, i educible CP- iola ing phase exis s,
non- emo able ia axion acuum choice.
The e o e, na u al solu ion o s ong CP p oblem in unied uni e se equi alen o opological
class
[K]
i iali y,  h cons ain on pa ame e amily
p
.
3.6 G a i a ional Wa e Dispe sion and
ℓcell
Obse a ional Uppe Bound
Theo em 7
(3.6: E en-O de G a i a ional Wa e Dispe sion and La ice Spacing Uppe Bound)
.
In g a i yQCA model, assume g a i a ional wa e dispe sion ela ion
ω2=c2k2h1 + X
n≥1
β2n(kℓcell)2ni,
odd-o de e ms excluded by NullModula and unied causalen opy consis ency. Then g oup
eloci y de ia ion
g
c−1≃X
n≥1
(2n+ 1)β2n(kℓcell)2n.
8
Using cons ain om GW170817/GRB170817A and subsequen e en s
| g/c −1|≲10−15
holding in hund ed Hz band, ob ain uppe bound on lowes -o de coecien
|β2|(kℓcell)2≲10−15,
hus unde
β2
na u alness assump ion p o iding uppe bound on
ℓcell
, e.g.,
ℓcell ≲10−30 m
o de .
Simul aneously his cons ain oge he wi h Theo em 3.1 black hole en opy la ice spacing lowe
bound p o ide o e lapping in e al, apping
ℓcell
wi hin ni e scale window.
This cons i u es six h cons ain on
(ℓcell, β2n)
.
3.7 Unied Solu ion Space Non-Emp iness
Theo em 8
(3.7: Common Solu ion Space Non-Emp y o Six Cons ain s)
.
Unde Sec ion 2
assump ions and echnical condi ions, pa ame e poin amily class exis s
p⋆=ℓ⋆
cell,∆ ⋆, d⋆
cell,H⋆
cell, κ⋆(ω),[K]⋆, β⋆
2n,
ETH da a
⋆,
a o da a
⋆,
making all cons ain s in Theo ems 3.13.6 simul aneously hold. In o he wo ds, unied ma ix
QCA uni e se objec class exis s whose black hole en opy, cosmological cons an , neu ino mass
and a o mixing, ETH, s ong CP, and g a i a ional wa e dispe sion mu ually compa ible on same
s uc u al pa ame e se .
This heo em uni o mly ew i es six unied physics p oblems om six independen p oblems o
consis ency condi ion on ni e-dimensional pa ame e space, p o ing hei common solu ion space
non-emp y.
4 P oo s
This sec ion p o ides p oo ideas o each heo em, lea ing echnical de ails o appendices.
4.1 Theo em 3.1: Black Hole En opy and Ho izon Cells
P oo di ides in o h ee s eps.
1.
Ho izon band la ice embedding and a ea coun ing
In globally hype bolic Lo en zian geome y selec black hole ho izon sec ion
ΣH
, cons uc ap-
p oxima ely equidis an la ice embedding on i , making la ice poin numbe
NH
sa is y
NH=A(ΣH)/ℓ2
cell +O(A0)
. Fo smoo h sec ions his cons uc ion is s anda d, e o e m om
cu a u e and bounda y eec s, con ollable ia local coo dina es and olume compa ison heo em.
2.
Typical en anglemen en opy and local dimension
On ho izon Hilbe space
HH≃ H⊗NH
g a
, conside ypical pu e s a es unde ene gy shell cons ain ,
using Le y concen a ion and Haa andom s a e en anglemen en opy es ima e ob ains
E[Sen ] = NHlog de +O(1),
whe e
de ≤dg a
. This conclusion consis en wi h exis ing esul s on andom pu e s a e en an-
glemen en opy.
3.
Ma ching wi h Bekens einHawking a ea law
Requi ing
Sen (ΣH) = A/(4G) + O(A0)
, compa ing leading e m ob ains
log de /ℓ2
cell = 1/(4G)
. Necessi y om coecien ma ching in a ea law; suciency ensu ed by
en anglemen en opy ypicali y and known ela ion be ween gene alized en opyEins ein equa-
ions.
9
condensed ma e analog sys ems, con inuously con ac unied solu ion space ollowing g a i a-
ional wa e and cosmological obse a ion p og ess.
Acknowledgemen s, Code A ailabili y
Au ho s hank ela ed li e a u e and communi y o sys ema ic esea ch in black hole en opy,
cosmological cons an , neu inos, ETH, s ong CP p oblem, and g a i a ional wa e dispe sion, p o-
iding backg ound and e e ence o his pape . Nume ical p o o ype and QCA simula ion o unied
ma ixQCA uni e se amewo k desc ibed in his pape implemen able on gene al quan um simula-
ion pla o ms and enso ne wo k lib a ies, code s uc u e simple, bu no published accompanying
his a icle.
Re e ences
[1] J. D. Bekens ein, Black Holes and En opy, Physical Re iew D
7
, 23332346 (1973).
[2] S. W. Hawking, Pa icle C ea ion by Black Holes, Communica ions in Ma hema ical Physics
43
, 199220 (1975).
[3] Y. Zhang, Black Hole En opy: Mic oscopic s. Mac oscopic, lec u e no es, Uni e si y o
Science and Technology o China.
[4] S. Weinbe g, The Cosmological Cons an P oblem, Re iews o Mode n Physics
61
, 123
(1989).
[5] A. Y. Kamenshchik, A. T onconi, G. Ven u i, Vacuum Ene gy and Spec al Func ion Sum
Rules, Physical Re iew D
75
, 083514 (2007).
[6] G. E. Volo ik, On Spec um o Vacuum Ene gy, a Xi :0801.2714 (2008).
[7] Pa icle Da a G oup, Neu ino Masses, Mixing, and Oscilla ions, in Re iew o Pa icle
Physics (2020).
[8] C. Gigan i, S. La ignac, M. Zi o, Neu ino Oscilla ions: The Rise o he PMNS Pa adigm,
P og ess in Pa icle and Nuclea Physics
98
, 154 (2018).
[9] K. Abe e al. (T2K Collabo a ion) and ela ed oscilla ion expe imen s, s anda d global-
summa ies as in PDG.
[10] L. D'Alessio, Y. Ka i, A. Polko niko , M. Rigol, F om Quan um Chaos and Eigens a e
The maliza ion o S a is ical Mechanics and The modynamics, Ad ances in Physics
65
, 239362
(2016).
[11] M. Rigol, V. Dunjko, M. Olshanii, The maliza ion and i s Mechanism o Gene ic Isola ed
Quan um Sys ems, Na u e
452
, 854858 (2008).
[12] I. M. D. A. Nassa , Re iew o he Eigens a e The maliza ion Hypo hesis, g adua ion
p ojec epo , Zewail Ci y (2024).
[13] J. E. Kim, A Re iew on Axions and he S ong CP P oblem, AIP Con e ence P oceedings
1200
, 8393 (2010).
[14] J. E. Kim, G. Ca osi, Axions and he S ong CP P oblem, Re iews o Mode n Physics
82
, 557602 (2010).
[15] R. Poggiani, GW170817: A Sho Re iew o he Fi s Mul imessenge G a i a ional Wa e
E en , Galaxies
13
, 112 (2025).
[16] J. H. Rao e al., Simula ion S udy on Cons aining GW P opaga ion Speed wi h Time
Delay Be ween GW and EM Signals, a Xi :2405.13314 (2024).
[17] K. V. Bayandin, Causal Disc e e Field Theo y o Quan um G a i y, a Xi :2001.10819
(2020).
16

[18] L. S. T ezzini, G. M. D'A iano, Reno malisa ion o Quan um Cellula Au oma a, Quan um
9
, 1756 (2025).
[19] Comp ehensi e QCA, causal se s, and disc e e g a i y ela ed wo ks and e iews, see disc e e
quan um g a i y and quan um in o ma ion in e disciplina y li e a u e compila ion.
A Black Hole En opy and G a i yQCA La ice Spacing Con-
s ain De ails
Assume ho izon band subla ice
ΓH⊂Λ
embedding sa ises
NH=A/ℓ2
cell +O(A0)
. Ho izon Hilbe space
HH≃ H⊗NH
g a
, conside ypical pu e s a e dis ibu ion unde ene gy shell cons ain , using clas-
sic esul o andom pu e s a e en anglemen en opy unde Haa measu e, ob ains c oss-ho izon
en anglemen en opy expec a ion
E[Sen ] = NHlog de +O(1), de ≤dg a .
E o e m
O(1)
con olled by local ene gy cons ain and ni e size co ec ion, no g owing
wi h a ea
A
.
On o he hand, ela ion be ween gene alized en opyEins ein equa ions indica es, in g a i a-
ional back- eac ion equilib ium s a e, gene alized en opy
Sgen =A
4G+Sou
a ia ion equi alen o Eins ein equa ions, leading a ea e m
A/(4G)
. In QCA uni e se equi ing
Sen (ΣH) = A/(4G) + O(A0)
, hen leading e m compa ison ob ains
log de
ℓ2
cell
=1
4G,
i.e.
ℓ2
cell = 4Glog de
. When
de ∼ O(1

10)
, his ela ion xes
ℓcell
a Planck leng h o de .
B Cosmological Cons an Windowed Sum Rule Taube ian P oo
Ou line
Conside hea ke nel die ence
∆K(s) = Z∞
0
e−sω2Θ′(ω) dω,
whe e
Θ′(ω) = ∆ρω(ω) = −ξ′(ω)
. In oduce loga i hmic window ke nel
W(ln(ω/µ))
, le i s
Mellin ans o m sa is y
Z∞
0
ω2nW(ln(ω/µ)) d ln ω= 0, n = 0,1.
Using MellinLaplace co espondence, can p o e in
s→0+
,
µ∼s−1/2
limi , small
s
hea ke nel
ni e pa equi alen o windowed spec al in eg al
ZΘ′(ω)W(ln(ω/µ)) d ln ω
17
he eby ew i ing acuum ene gy UV di e gence as windowed spec al in eg al. I QCA band
s uc u e sa ises sum ule in UV egion
REUV
0E2∆ρ(E) dE= 0
, hen
s−2
and
s−1
e ms in small
s
hea ke nel expansion anish, lea ing
only ni e e m ela ed o IR pa , co esponding o
Λe ∼E4
IR
.
C Axioma ically Chao ic QCA and ETH Design Es ima e
Assume on ni e egion
Ω
, QCA es ic ion
UΩ
iewable as dep h
d
local andom ci cui , local
ga e composi ion gene a ing app oxima e Haa dis ibu ion a e se e al laye s. In Hilbe space
dimension
D∼exp(s|Ω|)
, Haa andom uni a y ma ix elemen s a is ics gi e:
* Diagonal elemen s:
E[⟨ψn|OX|ψn⟩] = ⟨OX⟩mic o
, a iance
∼ O(D−1)
;
* O-diagonal elemen s:
E[|⟨ψm|OX|ψn⟩|2]∼ O(D−1)
.
Le y concen a ion inequali y gi es
P⟨ψn|OX|ψn⟩−⟨OX⟩mic o> ϵ≤Cexp(−cϵ2D),
since
D∼exp(s|Ω|)
, his p obabili y exponen ially decays wi h egion olume. Ex end Haa
andom case o app oxima e uni a y design ci cui , ob aining ETH o m in Theo em 3.4.
D Rela i e Cohomology Class
[K] = 0
and S ong CP Supp ession
Sca e ing de e minan line bundle squa e oo
L1/2
de
wis class
[K]∈H2(Y, ∂Y ;Z2)
unde s andable
as
Z2
single- aluedness obs uc ion on NullModula double co e . When
[K]= 0
, closed pa ame e
loop
γ⊂X◦
exis s, making squa e oo choice unde go sign ip along
γ
, co esponding o ce ain
CP-odd phase non- emo able ia local eld edeni ion.
Embedding QCD
θ
- e m and Yukawa phase in o
L1/2
de
be coo dina e, i
[KQCD]=0
, global
smoo h squa e oo choice exis s, can abso b physical
¯
θ
ia global phase edeni ion, making s ong
CP iola ion disappea ; i
[KQCD]= 0
, no such possibili y, axion eld also canno comple ely
elimina e CP iola ion ia local po en ial minimum. The e o e, iew
[K] = 0
as unied uni e se
consis ency condi ion, s ong CP p oblem ew i en as opological backg ound choice p oblem.
E G a i yQCA Dispe sion and LIGO/Vi go Cons ain Es ima-
ion
Conside dispe sion ela ion
ω2=c2k21 + β2(kℓcell)2,
g oup eloci y
g=∂ω
∂k ≃ch1 + 3
2β2(kℓcell)2i,
hus

g
c−1≃3
2|β2|(kℓcell)2.
In GW170817 equency band,
∼100 Hz
,
k∼2π /c ∼10−6m−1
, obse a ion p o ides
18
| g/c −1|≲10−15
. Subs i u ing ob ains
|β2|ℓ2
cell ≲O(10−3) m2.
I assuming
β2∼ O(1)
, hen
ℓcell ≲10−1.5m
his uppe bound nea ly uncons ained on cos-
mological scale. Howe e combined wi h highe - equency g a i a ional wa es o o he high-ene gy
as ophysical p ocess cons ain s on
k4
ype dispe sion pa ame e (usually exp essed as eec i e
mass o cu o scale
M∗
) [15,16], can comp ess
ℓcell
uppe bound o nea Planck scale se e al o de s
o magni ude abo e, he eby o ming non-emp y o e lapping window wi h lowe bound om black
hole en opy.
19