Equivalence Between Physical Universe and Matrix Universe: Causal Manifolds, Boundary Time Geometry, and Scattering Matrix Universe THE-MATRIX

Equi alence Be ween Physical Uni e se and Ma ix Uni e se:
Causal Mani olds, Bounda y Time Geome y, and Sca e ing
Ma ix Uni e se THE-MATRIX
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
Building on he unied amewo k o causal mani olds, axioma ic causal s uc u e, unied
ime scale and bounda y ime geome y, NullModula double co e , and in o ma ion geome ic
a ia ional p inciple, his pape in oduces and cha ac e izes a new on ological objec : sca e ing
ma ix uni e se THE-MATRIX, and p o es i s ca ego ical equi alence o physical uni e ses
sa is ying specic axiom amilies.
On one hand, we model he physical uni e se as a causal mani old objec wi h causal pa ial
o de , bounda y obse able algeb a, modula ow, gene alized en opy, and unied ime scale
mo he ule
Ugeo = (M, g, ≺,A∂, ω∂, Sgen, κ),
whe e unied ime scale is gi en by scale iden i y
κ(ω) = φ′(ω)/π =ρ el(ω) = (2π)−1 Q(ω),
wi h
φ
o al sca e ing hal -phase,
Q(ω)
Wigne Smi h ime-delay ma ix,
ρ el
ela i e densi y
o s a es, unied ia Bi manK en o mula and spec al shi unc ion.
On he o he hand, we dene THE-MATRIX uni e se as a one-pa ame e uni a y amily
S(ω)
ac ing on di ec sum Hilbe space
H=M
D∈D
HD.
I s block ma ix spa si y pa e n encodes causal pa ial o de ; diagonal block sca e ing phase
and ime delay ealize unied ime scale; block s uc u e and sel - e e en ial closed loops ca y
NullModula double co e and
Z2
opological sec o . Each small causal diamond
D
co esponds
o a bounda y sca e ing block
SDD(ω)
, whose local gene alized en opy ex emal condi ion and
second-o de non-nega i i y a e equi alen o local Eins ein equa ions and hei s abili y unde
in o ma ion geome ic a ia ional p inciple.
A ca ego ical le el, we cons uc geome ic uni e se ca ego y
Unigeo
and ma ix uni e se
ca ego y
Unima
, wi h mo phisms p ese ing causal, scale, and en opy s uc u es, and p esen
encoding unc o
F:Unigeo →Unima
and decoding unc o
G:Unima →Unigeo.
Unde axioms including global hype bolici y, local spec al econs uc abili y, ni e-o de
Eule Maclau in and Poisson e o discipline, NullModula double co e comple eness, and
1
gene alized en opy a ia ional comple eness, we p o e
F
and
G
a e quasi-in e se, yielding
uni e se ca ego y equi alence
Unigeo ≃Unima .
Building on his, we o malize obse e s as comp essions and eadou ope a o s on ma ix
uni e se, showing ha he wo ld seen by specic obse e s is a c oss-sec ion o THE-MATRIX,
while mul i-obse e consensus p oblems can be o mula ed as geome ic and in o ma ional con-
sis ency condi ions be ween die en c oss-sec ions, in e acing wi h causal ne wo ks, axioma ic
ela i e en opy, and modula ow heo y.
Keywo ds
Causal mani olds; Unied ime scale; Bounda y ime geome y; Wigne Smi h ime delay; Sca -
e ing ma ix; Spec al shi unc ion; NullModula double co e ; Gene alized en opy; Rela i e
en opy; Ma ix uni e se; Ca ego ical equi alence
1 In oduc ion & His o ical Con ex
Gene al ela i i y and quan um eld heo y ypically adop ligh cones on ou -dimensional Lo en z
mani olds and local elds as undamen al s uc u es; quan um many-body and quan um in o ma ion
heo y end owa d ma ices, ope a o a ays, and ne wo ks as p ima y language. The b idge be-
ween hem adi ionally elies on spec al heo y, sca e ing heo y, and ope a o algeb as: Bi man
K en o mula links sca e ing ma ix de e minan wi h spec al shi unc ion; Wigne Smi h ime
delay in e p e s sca e ing phase g adien as ope a o ized scale o  ime delay; Tomi aTakesaki
modula heo y and A aki ela i e en opy p o ide unied s uc u e among ime, empe a u e, and
in o ma ion mono onici y a on Neumann algeb a le el; Malamen and HawkingKingMcCa hy
o malized he idea ha causal s uc u e de e mines space ime opology and con o mal class.
These de elopmen s join ly poin o a na u al ques ion: can we iew physical uni e se as some
gian sca e ing ma ix uni e se THE-MATRIX, such ha geome iccausal pic u e and ma ix
ope a o pic u e a e equi alen ? This concep ion has shown agmen a y signs ac oss mul iple
esea ch lines:

Sca e ing geome y and g a i y
: Unde app op ia e bounda y condi ions, GHY bound-
a y e m and B ownYo k ene gy can be es a ed ia bounda y sca e ing and spec al shi
unc ion;

Modula ow and he mal ime
: ConnesRo elli he mal ime hypo hesis iews ime as
modula ow pa ame e induced by s a ealgeb a pai ;

Causal se s and disc e e space ime
: App oxima e Lo en z mani old by pa ially o de ed
se s, u ilizing Malamen - ype heo ems o econs uc opology and con o mal class o me ic;

Black hole he modynamics and dynamical s abili y
: HollandsWald e o mula e s a-
bili y p oblem as posi i i y o canonical ene gy, whose second-o de a ia ion is closely e-
la ed o Hessian o gene alized en opy.
On he o he hand, physical sys ems such as sca e ing ne wo ks, quan um g aphs, Floque -
d i en la ices p o ide na u al ealiza ions o la ge-scale uni a y block ma ices, making ma ix
uni e se po en ially enginee ing- ealizable.
Building on exis ing wo k, his pape p oposes and sys ema izes he ollowing pic u e:
2
1. Wi h small causal diamonds and hei bounda y obse able algeb as as local uni s, comp ess
physical uni e se in o causal mani old objec
Ugeo
wi h scale mo he ule and gene alized
en opy s uc u e;
2. Cha ac e ize ma ix uni e se THE-MATRIX by uni a y block ma ix amily
S(ω)
on di ec
sum Hilbe space, whose spa si y pa e n encodes causal pa ial o de , diagonal blocks encode
bounda y ime geome y and gene alized en opy;
3. A ca ego ical le el, cons uc encoding unc o
F
and decoding unc o
G
, p o ing
Unigeo ≃
Unima
unde app op ia e axioms.
This gi es igo ous ma hema ical meaning o physical uni e se = ma ix uni e se THE-MATRIX,
p o iding s uc u al explana ion o unica ion among obse e consensus, causal ne wo ks, and op-
e a o ne wo ks.
2 Model & Assump ions
2.1 Geome ic Uni e se Model
Ugeo
Le
(M, g)
be a ou -dimensional, o ien able, ime-o ien able, globally hype bolic Lo en z mani old,
wi h causal ela ion deno ed
≺
. Fo each poin
p∈M
and sucien ly small scale pa ame e
ℓ > 0
,
dene small causal diamond
Dℓ(p) = I+(p−)∩I−(p+),
whe e
p±
a e displaced by
ℓ
along some p ope ime geodesic. Choose label se amily
D
and map
α7→ Dα⊂M
sa is ying:
1. Each
Dα
is some
Dℓ(p)
;
2.
{Dα}α∈D
co e s
M
and is locally ni e;
3. I
Dα∩Dβ=∅
, o e lap egion emains globally hype bolic.
Dene pa ial o de on
D
by
α⪯β⇐⇒ Dα⊂J−(Dβ).
Fo each
Dα
, endow bounda y
∂Dα
wi h on Neumann algeb a
A∂(Dα)
and ai h ul s a e
ωα
,
sa is ying inclusion
Dα⊂Dβ⇒ A∂(Dα)⊂ A∂(Dβ).
Fo each pai
(A∂(Dα), ωα)
, conside Tomi aTakesaki modula ow
{σ(α)
} ∈R
, whose gene a o
Kα
localizes o null slice o
∂Dα
in GNS ep esen a ion.
On each
Dα
bounda y conside xed-ene gy sca e ing p oblem, wi h sca e ing ma ix
Sα(ω)
and Wigne Smi h ime delay
Qα(ω) = −iSα(ω)†∂ωSα(ω).
Bi manK en o mula gi es spec al shi unc ion
ξα(ω)
and sca e ing de e minan
de Sα(ω) = exp(−2πiξα(ω)),
wi h spec al shi unc ion de i a i e being ela i e densi y o s a es
ρ el,α(ω) = ξ′
α(ω)
.
3
Unied ime scale mo he ule dened as
κα(ω) = φ′
α(ω)/π =ρ el,α(ω) = (2π)−1 Qα(ω),
whe e
φα(ω) = πξα(ω)
is o al sca e ing hal -phase. T ace dened unde ni e-o de Eule 
Maclau in and Poisson e o discipline, ensu ing singula i y non-g ow h.
Gene alized en opy akes o m
Sgen,α =Aα/(4Gℏ) + S en
ou ,α +SUV
c ,α −ΛVα/(8πGTα),
whe e
Aα
is wais su ace a ea,
S en
ou ,α
eno malized ex e io en opy,
SUV
c ,α
local coun e e m,
Vα
diamond olume,
Tα
modula o Un uh empe a u e.
Axiom 1
(IGVP (Geome ic Ve sion))
.
In small-scale limi
ℓ→0
, o each
p∈M
and nea by
diamond amily
Dℓ(p)
:
1. Fo any a ia ion sa is ying app op ia e bounda y condi ions,  s a ia ion
δSgen = 0
;
2. Second a ia ion denes non-nega i e quad a ic o m;
3. Limi and a e aging ope a ions commu e, allowing gene aliza ion o gene al s a e amilies.
Unde s anda d egula i y assump ions, his axiom is equi alen o local Eins ein equa ions and
posi i i y o HollandsWald canonical ene gy.
Deni ion 2
(Geome ic Uni e se Objec )
.
A geome ic uni e se is a se en- uple
Ugeo = (M, g, ≺,{A∂(Dα), ωα}α∈D,{κα}α∈D,{Sgen,α}α∈D),
sa is ying abo e geome ic, algeb aic, scale, and IGVP axioms.
2.2 Ma ix Uni e se Model
Uma
Take locally ni e pa ially o de ed se
(D,⪯)
, wi h each elemen 's pas and u u e cones ni e, and
scale map
ℓ:D → (0, ℓ0]
. Fo each
α∈ D
ake sepa able Hilbe space
Hα
, dening di ec sum
H=M
α∈D
Hα.
Dene s ongly con inuous map
ω7→ S(ω)∈ U(H)
such ha o each
ω
,
S(ω)
has block ma ix
o m
Sαβ(ω) : Hβ→ Hα
unde di ec sum decomposi ion, sa is ying uni a i y condi ions.
Axiom 3
(Causal Spa si y)
.
I
Sαβ(ω)= 0
, hen
α⪯β
.
Fo each
α
, dene diagonal block
Sαα(ω)
and
Qα(ω) = −iSαα(ω)†∂ωSαα(ω),
se ing
κα(ω) = (2π)−1 Qα(ω).
Requi e exis ence o sca e ing hal -phase
φα(ω)
and ela i e densi y o s a es
ρ el,α(ω)
such ha
scale iden i y
κα(ω) = φ′
α(ω)/π =ρ el,α(ω) = (2π)−1 Qα(ω)
holds wi h e o con olled by ni e-o de Eule Maclau in and Poisson discipline.
A each
α
assume exis ence o NullModula double co e decomposi ion o modula Hamil onian
Kα
and
Z2
ledge
χα
; p oduc o e closed causal diamond chains yields opological sec o .
Ma ix uni e se u he ca ies gene alized en opy unc ion amily
Sgen,α
cons uc ed om block
ma ix spec um, sa is ying ma ix e sion o IGVP axiom.
4
Deni ion 4
(Ma ix Uni e se Objec )
.
A ma ix uni e se is a  e- uple
Uma = (D,⪯,{Hα}α∈D,S(ω),{κα, χα, Sgen,α}α∈D),
sa is ying abo e causal spa si y, scale, NullModula , and IGVP condi ions.
3 Main Resul s (Theo ems and Alignmen s)
This sec ion s a es main esul s a ca ego ical le el, p esen ing equi alence ela ion be ween geo-
me ic and ma ix uni e ses.
3.1 Uni e se Ca ego ies and Mo phisms
Deni ion 5
(Geome ic Uni e se Ca ego y
Unigeo
)
.
Objec s a e all geome ic uni e ses
Ugeo
sa -
is ying abo e axioms. Mo phism
:Ugeo →U′
geo
consis s o :
1. Causal homeomo phism
M: (M, g, ≺)→(M′, g′,≺′)
;
2. Isomo phism
D → D′
on small causal diamond co e ing indices sa is ying
(Dα) = D′
(α)
;
3. Fo each
α
,
∗
-isomo phism
Φα:A∂(Dα)→ A∂(D′
(α))
o on Neumann algeb as, consis en
wi h s a es:
ω′
(α)◦Φα=ωα
;
4. Scale densi y and gene alized en opy p ese ed unde
:
κ′
(α)=κα◦ −1
,
S′
gen, (α)=Sgen,α
.
Deni ion 6
(Ma ix Uni e se Ca ego y
Unima
)
.
Objec s a e all ma ix uni e ses
Uma
sa is ying
abo e axioms. Mo phism
Ψ : Uma →U′
ma
consis s o pa ial o de isomo phism
ψ:D → D′
and Hilbe space uni a y ope a o
U:H → H′
such ha
US(ω)U†=S′(ω), U(Hα) = H′
ψ(α),
p ese ing
{κα, χα, Sgen,α}
da a.
3.2 Encoding and Decoding Func o s
Deni ion 7
(Encoding Func o
F:Unigeo →Unima
)
.
Fo objec
Ugeo
:
1. Take small causal diamond co e ing index se
D
wi h pa ial o de
⪯
;
2. Fo each
α
, ake GNS Hilbe space
Hα
o bounda y sca e ing channel space;
3. On di ec sum
H=LαHα
cons uc global sca e ing ope a o
S(ω)
whose block ma ix
Sαβ(ω)
is de e mined by geome ic uni e se's bounda y condi ions, p opaga ion, and eec ion
s uc u e; causali y ensu es spa si y pa e n;
5

4. Diagonal blocks
Sαα(ω)
wi h gene alized en opy and NullModula da a a e gi en by geo-
me ic uni e se axioms, di ec ly assigned o ma ix uni e se.
Ob ain ma ix uni e se
F(Ugeo)
.
Fo mo phism
:Ugeo →U′
geo
, GNS uni e sal p ope y and sca e ing cons uc ion yield uni a y
ope a o
U :H → H′
and index isomo phism
D → D′
, hus ob aining mo phism
F( )
, making
F
a unc o .
Deni ion 8
(Decoding Func o
G:Unima →Unigeo
)
.
Fo objec
Uma
:
1. View
(D,⪯)
as abs ac causal ne wo k, econs uc ing opology and con o mal s uc u e ia
Alexand o opology and Malamen HawkingKingMcCa hy ype heo em;
2. Combining high- and low- equency beha io o scale densi y
κα(ω)
, use spec al geome ic
me hods o econs uc bounda y spec al iple and me ic agmen s om local sca e ing
blocks
Sαα(ω)
, de e mining con o mal ac o and p ope ime scale o me ic;
3. Cons uc gene alized en opy
Sgen,α
om block ma ix spec um, de i ing Eins ein equa ions
in small diamond limi ia IGVP axiom, ob aining Lo en z mani old
(M, g)
and i s causal
s uc u e
≺
;
4. Cons uc bounda y obse able algeb a
A∂(Dα)
and s a e
ωα
om block ma ix in-ou s uc-
u e; econs uc modula ow and NullModula double co e om
κα
and
χα
.
Ob ain geome ic uni e se
G(Uma )
.
Fo mo phism
Ψ : Uma →U′
ma
, pa ial o de isomo phism and Hilbe space uni a y ope a o
induce causal homeomo phism and bounda y algeb a isomo phism, yielding
G(Ψ)
, making
G
a
unc o .
3.3 Main Equi alence Theo em
To s a e main esul , in oduce he ollowing mu ual econs uc abili y axiom.
Axiom 9
(Geome icMa ix Mu ual Recons uc abili y)
.
1. Fo any
Ugeo ∈Unigeo
, encoding
F(Ugeo)
sa ises ma ix uni e se axioms, p ese ing all opological, scale, and gene alized
en opy in o ma ion;
2. Fo any
Uma ∈Unima
, decoding
G(Uma )
sa ises geome ic uni e se axioms, wi h econ-
s uc ed causal mani old and bounda y ime geome y unique up o isomo phism;
3. All spec algeome ic econs uc ion uses only ni e-o de Eule Maclau in and Poisson ex-
pansion, sa is ying singula i y non-g ow h p inciple;
4. Scale unc ion
ℓ(α)
o index se
D
is sucien ly dense so small diamond limi s and Radon- ype
closu es a e well-dened;
5.
Z2
ledge
χα
and NullModula da a comple ely eco d opological sec o s, allowing comple e
econs uc ion o ma ix uni e se opological s uc u e a geome ic le el.
Theo em 10
(Ca ego ical Equi alence o Geome ic and Ma ix Uni e ses)
.
Unde abo e mu ual
econs uc abili y and egula i y axioms, encoding unc o
F:Unigeo →Unima
and decoding unc o
G:Unima →Unigeo
a e quasi-in e se, yielding ca ego ical equi alence
Unigeo ≃Unima .
6
4 P oo s
This sec ion p o ides p oo ou line o main equi alence heo em, wi h mo e echnical a gumen s in
appendices.
4.1 Fullness and Fai h ulness o
F
P oposi ion 11
(Fullness)
.
I wo geome ic uni e ses
Ugeo, U′
geo
sa is y
F(Ugeo)∼
=F(U′
geo),
hen
Ugeo ∼
=U′
geo
.
P oo ou line
:
1.
Causal ne wo k isomo phism
: Ma ix uni e se nonze o block spa si y pa e n de e mines
abs ac causal ne wo k
(D,⪯)
. Ma ix uni e se isomo phism implies pa ially o de ed se
isomo phism, hence isomo phism o small causal diamond co e ing indices and hei pa ial
o de s o wo geome ic uni e ses;
2.
Local geome ic econs uc ion
: Fo each
α
, block ma ix diagonal elemen
Sαα(ω)
sca -
e ing spec um coincides; combined wi h Bi manK en o mula and spec al geome y he-
o y, uniquely econs uc s bounda y spec al iple and con o mal class o local me ic ag-
men
g|Dα
;
3.
Scale and olume in o ma ion
: High- equency beha io o scale densi y
κα(ω)
gi es
coecien s o bounda y Di ac spec um coun ing unc ion, de e mining wais su ace a ea
and small olume quan i ies; combining causal s uc u e and olume in o ma ion, Malamen 
HawkingKingMcCa hy ype heo em econs uc s con o mal ac o o me ic;
4.
IGVP laye cons ain
: Equi alence o gene alized en opy and i s a ia ion ensu es con-
sis ency o Eins ein equa ions and ma e s essene gy enso , excluding esidual deg ees o
eedom;
5.
Gluing uniqueness
: Sca e ing ma ix and en opy da a consis ency in o e lap egions en-
su es unique gluing o me ic and algeb a, yielding global causal homeomo phism and bound-
a y algeb a isomo phism.
The e o e
Ugeo
and
U′
geo
a e isomo phic in
Unigeo
.
P oposi ion 12
(Fai h ulness)
.
I wo mo phisms be ween geome ic uni e ses
, g :Ugeo →U′
geo
sa is y
F( ) = F(g)
, hen
=g
.
P oo ou line
:
1.
F( ) = F(g)
implies hei uni a y ealiza ions
U
and
Ug
on
H
coincide, wi h consis en
ac ion on index se ;
2. GNS ep esen a ion uni e sal p ope y ensu es on Neumann algeb a and s a e isomo phisms
comple ely de e mined by co esponding uni a y;
3. Thus geome ic and algeb aic le el mo phisms mus coincide, i.e.,
=g
.
Hence
F
is ully ai h ul.
7
4.2
G◦F≃idUnigeo
Fo any
Ugeo
,  s encode o ob ain
F(Ugeo)
, hen decode o ob ain
G(F(Ugeo))
. By cons uc ion:
1. Abs ac causal ne wo k
(D,⪯)
isomo phic o o iginal small causal diamond co e ing;
2. Local sca e ing blocks
Sαα(ω)
and scale densi y
κα(ω)
di ec ly gi en by geome ic uni e se;
3. Decoding p ocess, pe mu ual econs uc abili y axiom, necessa ily e u ns o o iginal
(M, g, ≺
)
and bounda y ime geome y, up o causal homeomo phism and algeb as a e uni a y iso-
mo phism.
Collec hese isomo phisms in o na u al ans o ma ion
η:G◦F⇒idUnigeo
.
4.3
F◦G≃idUnima
Fo any
Uma
,  s decode o ob ain
G(Uma )
, hen encode o ob ain
F(G(Uma ))
. Mu ual econ-
s uc abili y axiom ensu es:
1.
Causal ne wo k and opology
: Small causal diamond co e ing o econs uc ed
(M, g, ≺)
isomo phic o o iginal index se
D
;
2.
Local sca e ing blocks and scale
:
Sαα(ω)
and
κα(ω)
econs uc ed om geome ic uni-
e se coincide wi h o iginal ma ix uni e se;
3.
O-diagonal blocks
uniquely de e mined by p opaga ion pa hs and causal s uc u e; a e
encoding, global
S(ω)
is uni a ily equi alen o o iginal ma ix uni e se.
Thus he e exis s na u al ans o ma ion
ϵ:F◦G⇒idUnima
.
4.4 Na u ali y o Equi alence
Fo any mo phism
:Ugeo →U′
geo
, encoding-decoding and na u al isomo phisms sa is y
ηU′
geo ◦G(F( )) = ◦ηUgeo .
Simila ly o any ma ix uni e se mo phism
Ψ
,
ϵU′
ma ◦F(G(Ψ)) = Ψ ◦ϵUma .
This comple es p oo o ca ego ical equi alence.
5 Model Applica ions
This sec ion shows how his equi alence amewo k ew i es obse e s, consensus, and NullModula
double co e s uc u es.
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5.1 Obse e s as Ma ix Comp ession and Readou
In geome ic uni e se, an obse e can be abs ac ed as mul i-componen objec
Oi= (Ci,≺i,Λi,Ai, ωi,Mi, Ui, ui,{Cij}j),
whe e
Ci⊂M
is accessible causal domain,
Λi
esolu ion scale,
Ai
obse able algeb a,
ωi
s a e,
Mi
model amily,
Ui
upda e ope a o ,
ui
u ili y unc ion,
Cij
communica ion channels.
In ma ix uni e se, his co esponds o:
1. Index subse
Di⊂ D
ep esen ing obse e -accessible small causal diamonds;
2. Hilbe subspace
Hi=Lα∈DiHα
;
3. P ojec ion ope a o
Pi:H → Hi
;
4. Subma ix amily
S(i)(ω) = PiS(ω)P†
i
;
5. S a e amily and upda e ope a o s on
B(Hi)
desc ibing obse e 's belie and lea ning p ocess.
Obse e 's wo ld c oss-sec ion can be unde s ood as weigh ed sec ion
{(α, ωi,α)}α∈Di,
whose e olu ion is de e mined by subma ix
S(i)(ω)
and communica ion ope a o s wi h o he ob-
se e s.
5.2 Consensus and Conic
Mul i-obse e consensus can be decomposed in o h ee consis encies:
1.
Causal consis ency
: On o e lap egion
Di∩ Dj
, spa si y pa e n and pa ial o de mus be
compa ible:
S(i)
αβ(ω)= 0 ⇐⇒ S(j)
αβ(ω)= 0;
2.
Scale consis ency
: On common equency window and common diamonds, scale densi y and
loga i hmic de i a i e coincide:
κ(i)
α(ω) = κ(j)
α(ω),
co esponding o unied ime scale equi alence class;
3.
S a e and model consis ency
: S a es on common obse able algeb a con e ge o same xed
poin h ough i e a i e communica ion and Umegaki ela i e en opy mono onici y; model
amily in e sec ion con ac s o unique ue model unde da a accumula ion.
Ma ix uni e se p o ides ope a o ized exp ession o hese consis ency condi ions: all obse e
c oss-sec ions
S(i)
a e comp essions o same THE-MATRIX; consensus exis ence equi alen o ex-
is ence o global ma ix uni e se
Uma
and p ojec ion amily
{Pi}
such ha all c oss-sec ions and
comp ession condi ions a e compa ible.
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C.3 Second Va ia ion and Canonical Ene gy
Second a ia ion
d2
dλ2Sgen,αλ=0
in ma ix con ex can be exp essed as quad a ic o m on me ic and ma e eld pe u ba ions
Qα[δg, δϕ].
Using con exi y and mono onici y o A aki ela i e en opy and HollandsWald canonical ene gy
cons uc ion, can ela e
Qα
o second-o de a ia ion o canonical ene gy
E
:
Qα[δg, δϕ]≥0⇐⇒ E[δg, δϕ]≥0.
This p o ides consis ency be ween IGVP axiom second-o de laye and dynamical s abili y,
ensu ing small pe u ba ions on ma ix uni e se do no igge nega i e canonical ene gy modes,
p o iding in o ma ion geome ic c i e ion o THE-MATRIX s abili y.
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