Boundary as Unified Stage: Variational Completeness, Time Scale, Topological Branching

Bounda y as Unied S age:
Va ia ional Comple eness, Time Scale, Topological
B anching
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
F om  s p inciples, ele a e bounda y om passi e geome ic appendage o
unied physical s age. P opose axioma ic amewo k wi h bounda y as undamen al
objec , gluing h ee seemingly sepa a e s uc u esGibbonsHawkingYo k (GHY)
bounda y e m and B ownYo k quasilocal quan i ies in g a i a ional a ia ion,
spec al shi unc ion and Wigne Smi h ime delay in sca e ing heo y, modula
ow and ela i e en opy mono onici y in ope a o algeb asas die en p ojec ions
o same bounda y ime geome y.
Co e iewpoin : bounda y no me ely sepa a ing bulk domains bu comp ess-
ing bulk con inuous changes in o ni e measu able die ences (ene gy die ence,
ime die ence, opological class die ence, causal o ien a ion).
On geome ic a ia ional side, e iew and ene comple e a ia ional s uc u e
o Eins einHilbe GHYco ne null bounda y e ms, p o ing unde xed induced
me ic condi ion, equi ing well-dened a ia ional p inciple uniquely selec s  a ia-
ionally comple e bounda y geome y class, de i ing B ownYo k bounda y s ess
ene gy enso as bounda y eadou o bulk die ence.
On spec alsca e ing side, unde s anda d ace-class pe u ba ion assump-
ions, s a ing om Bi manK ein o mula, gi e scale iden i y ela ing o al sca -
e ing phase de i a i e, ela i e s a e densi y, Wigne Smi h ime delay ace, in-
e p e ing bounda y sca e ing phase iny a ia ions uni o mly as s a e numbe
changes and  esidence ime changes.
On ope a o algeb ain o ma ion side, in oduce modula ow unde gene al
bounda y obse able algeb a and ai h ul s a e con ex , w i e modula Hamil o-
nian as ene gy ow in eg al along null bounda y (o wedge bounda y), cha ac e -
ize bounda y ime a ow unidi ec ionali y using ela i e en opy mono onici y and
quan um ene gy condi ions.
Mo eo e , dene disc iminan bounda y and
Z2
b anch index in pa ame e
space, explaining a e excluding spec al anomaly and opological phase ansi ion
hype su aces, spin s uc u e dened by sca e ing ma ix squa e oo o ms Null
Modula double co e on pa ame e space. Double co e 's non- i iali y memo ized
by bounda y as spec al ow pa i y and in e sec ion numbe pa i y, comp essing
going a ound once con inuous de o ma ion in o disc e e opological class die -
ence.
1
Finally, gi e unied bounda y ime geome y deni ion: bounda y ca ies
geome icspec alin o ma ion opological da a se making (i) a ia ion well-dened,
(ii) ime scale iden i y holds, (iii) modula ow and gene alized en opy mono onic-
i y de e mine ime a ow, (i )
Z2
index dened on disc iminan bounda y gi es
opological b anching.
Main heo em: unde app op ia e assump ions, can selec unique (in ane
escaling sense) ime pa ame e on bounda y making hese ou s uc u e ime pa-
ame e s belong o same scale equi alence class, hus p ecisely o malizing bound-
a y gene a es die ence in o es able, compu able unied amewo k.
Keywo ds:
Bounda y Geome y; Va ia ional Comple eness; Spec al Shi ; Time Delay;
Modula Flow; Rela i e En opy; Topological Index;
Z2
B anching

1 In oduc ion
Bounda ies appea ubiqui ously in almos all co ne s o physics and ma hema ics: spa-
ial inni y and black hole ho izons in gene al ela i i y, Cauchy su aces and causal
diamond bounda ies in quan um eld heo y, ma e ial in e aces and opological de ec s
in condensed ma e , incoming/ou going asymp o ic bounda ies in sca e ing heo y, e en
disc iminan hype su aces o opological phase ansi ions in pa ame e space.
T adi ional ea men s mos ly iew bounda y as geome ic appendage o bulk do-
main: in eld heo y need o speci y bounda y condi ions, in geome y need o supple-
men bounda y e ms o co ec a ia ion, in sca e ing heo y bounda y me ely way o
imposing asymp o ic condi ions a inni y.
This pape a emp s o ad ance mo e adical iewpoin : in unied amewo k, **bound-
a y should be iewed as ue s age o physical s uc u e**. Bulk con inuous changes
only become measu able, compa able, op imizable objec s when ansla ed on bounda y
in o ni e-dimensional scale die ences.
Mo e specically:
•
On **geome ic a ia ional** le el, equi ing Eins einHilbe
ac ion a ia ion well-dened in bounda y case o ces us o in oduce GibbonsHawking
Yo k e m and co ne /null bounda y e ms on bounda y; B ownYo k quasilocal s ess
ene gy enso na u ally appea s as bounda y ledge o how much bulk geome y and
ma e dis ibu ion die .
•
On **spec alsca e ing** le el, Bi manK ein spec al shi unc ion and Wigne 
Smi h ime delay ansla e  iny phase changes wi h equency in o s a e densi y die -
ence and  esidence ime die ence; his s uc u e na u ally is bounda y s uc u e since
all sca e ing eadou s measu ed on bounda y (o a inni y).
•
On **ope a o algeb ain o ma ion** le el, Tomi aTakesaki modula heo y and
ela i e en opy mono onici y show: gi en bounda y obse able algeb a and ai h ul s a e,
can dene modula ow and i s gene a o (modula Hamil onian), o en w i able as en-
e gy ow in eg al along bounda y; ow pa ame e a e app op ia e no maliza ion in e -
p e able as in insic ime on bounda y.
•
On ** opologicalpa ame e space** le el, sca e ing ma ix spec um and phase
in pa ame e space o en ha e disc iminan hype su aces; excluding hese anomalous
poin s, spin s uc u e dened by squa e- oo sca e ing ma ix o ms
Z2
double co e on
pa ame e space, non- i iali y mani es ed on bounda y as ex a minus sign a e going
a ound once.
2
These seemingly sca e ed phenomena poin o common s uc u e: **bounda y e-
sponsible o comp essing in isible bulk changes in o isible die ence scales**. This
pape 's goal: s a ing om his in ui ion, cons uc igo ous axioma ic bounda y ame-
wo k; gi e heo em se ies uni ying a ia ional comple eness, ime scale iden i y, modula
ow ime a ow, opological b anching in o bounda y ime geome y con ex .

2 P elimina ies and No a ion
2.1 Geome y and Va ia ion
Le
(M, g)
be ou -dimensional Lo en zian mani old wi h bounda y
∂M
. On non-null
(spacelike o imelike) bounda y, deno e induced me ic
hab
, ou wa d no mal
na
, ex insic
cu a u e
Kab =hc
ahd
b∇cnd
, ace
K=habKab
.
Eins einHilbe ac ion dened as
SEH(g) = 1
16πG ZM
R(g)√−g d4x,
whe e
R
is scala cu a u e. Well-known: in bounda y case,
SEH
alone no well-dened
unde a ia ion xing
hab
; a ia ion p oduces bounda y e m depending on
δ(∂g)
. To
co ec , in oduce GibbonsHawkingYo k bounda y e m
SGHY(g) = ε
8πG Z∂M
Kp|h|d3x,
whe e
ε= +1
o spacelike bounda y,
ε=−1
o imelike bounda y. Fo co ne and null
bounda y cases, need in oduce addi ional co ne and null bounda y e ms; see Appendix
A.
2.2 Sca e ing Theo y and Time Delay
Le
H
be complex Hilbe space,
H0
and
H=H0+V
sel -adjoin ope a o s. Assume
pe u ba ion
V
is
H0
- ela i e ace-class making wa e ope a o s
W±= s
-
lim
→±∞ eiH e−iH0
exis and comple e; hen sca e ing ope a o dened as
S=W∗
+W−.
Unde app op ia e condi ions,
S
w i able in ene gy ep esen a ion as be decompo-
si ion
S=Z⊕
S(ω)dµ(ω),
whe e
S(ω)
is ni e-dimensional (o sepa able) sca e ing ma ix a ene gy
ω
. Dene
Wigne Smi h ime delay ope a o
Q(ω) = −i S(ω)†∂ωS(ω).
Bi manK ein spec al shi unc ion
ξ(λ)
sa ises
de S(λ) = exp(−2πi ξ(λ)).
Unde app op ia e egula i y assump ions, die en iable; de i a i e
ξ′(λ)
ela ed o s a e
densi y die ence. Adop no a ion K ein spec al shi densi y
ρ el(ω) = ξ′(ω)
.
3
2.3 Modula Theo y, Modula Flow, Rela i e En opy
Le
A
be
C∗
algeb a o on Neumann algeb a,
ω
ai h ul no mal s a e on i . GNS
cons uc ion gi es iple
(πω,Hω,Ωω)
whe e
Ωω
is cyclic ec o . Tomi a ope a o
Sω
's
pola decomposi ion p oduces modula ope a o
∆ω
and conjuga ion
Jω
. Modula ow
dened as
σω
(A)=∆i
ωA∆−i
ω, A ∈ A.
I sel -adjoin ope a o
Kω
exis s sa is ying
∆ω=e−Kω
, o mally
σω
(A) = eiKω Ae−iKω .
Fo wo s a es
ω, φ
, ela i e en opy dened as
S(ω∥φ) = (ρω(log ρω−log ρφ)).
Unde app op ia e gene ali y sa ises mono onici y: non-inc easing unde es ic ion o
subalgeb a o sub egion.
In ela i is ic QFT double cone/wedge egion cases, modula Hamil onian
Kω
o en
w i able as ene gymomen um enso in eg al along bounda y di ec ion, gi ing modula
ime geome ic meaning.
2.4 Spec al Flow and
Z2
Index
Le
{A } ∈[0,1]
be amily o sel -adjoin F edholm ope a o pa hs; spec al ow
S ({A })
dened as o ien ed numbe o eigen alues c ossing ze o. I only ca ing abou pa i y, dene
Z2
index
νZ2({A }) = (−1)S ({A })∈ {±1}.
On pa ame e space
X
, can connec spec al ow pa i y wi h loop in e sec ion numbe
pa i y o disc iminan se
D⊂X
, o ming
Z2
opological index. Will use his language
o desc ibe sca e ing squa e- oo b anch s uc u e.

3 Axioma ic Deni ion o Physical Bounda y Sys em
Gi e physical bounda y sys em deni ion adop ed in his pape , uni ying geome ic,
sca e ing, modula ow, opological da a in o same bounda y amewo k.
3.1 Bounda y Da a Quad uple
Deni ion 3.1
(Physical Bounda y Sys em)
.
Physical bounda y sys em consis s o
quad uple
B= (∂M, A∂, ω∂,S∂)
whe e:
1.
∂M
is codimension-one bounda y o ou -dimensional space ime
M
, equipped wi h
induced me ic
hab
and ex insic cu a u e
Kab
, plus possible co ne s and null shee
segmen s;
2.
A∂
is bounda y obse able algeb a associa ed wi h
∂M
(e.g., bounda y- es ic ed
eld ope a o algeb a o sca e ing channel algeb a);
4
3.
ω∂
is ai h ul no mal s a e on
A∂
, gi ing modula ow
σω∂
;
4.
S∂
is se o bounda y sca e ing and opological da a, including:
•
Sca e ing ma ix
S(ω)
in ene gy ep esen a ion;
•
F equency measu e compa ible wi h
A∂
;
•
Disc iminan
subse
D⊂X
on pa ame e space
X
and
Z2
index dened om i .
In conc e e models,
∂M
can be a icial bounda y o ni e egion, small causal dia-
mond bounda y, black hole ho izon, AdS asymp o ic bounda y, ma e ial in e ace, e en
disc iminan bounda y in pa ame e space in abs ac sense.
3.2 Fou Fundamen al Pos ula es
Pos ula e 1
(A1: Va ia ional Comple eness)
.
Exis s ac ion unc ional composed o bulk
and bounda y
S o =Sbulk[g, Φ] + Sbdy[h, K, Φ|∂M ],
whe e specic bounda y e m
Sbdy
(including GHY, co ne , null bounda y e ms) makes
unde a ia ion xing bounda y induced me ic and ma e bounda y da a
(hab,Φ|∂M )
,
 s -o de a ia ion o
S o
depends only on bulk a ia ion and equi alen o gi en eld
equa ions (e.g., Eins einma e equa ions).
Pos ula e 2
(A2: Scale Iden i y)
.
Exis s equency a iable
ω
and co esponding sca -
e ing ma ix
S(ω)
making ollowing scale iden i y hold:
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
whe e
φ(ω) = 1
2a g de S(ω)
,
ρ el(ω)
is ela i e s a e densi y,
Q(ω) = −iS(ω)†∂ωS(ω)
is
Wigne Smi h ime delay ma ix. Func ion dened om his
κ(ω) := φ′(ω)
π
called bounda y ime scale densi y.
Pos ula e 3
(A3: Modula Flow O ien a ion and Time A ow)
.
Modula ow
σω∂
gen-
e a o
K∂
w i able as in eg al o ene gymomen um enso p ojec ion along bounda y
K∂=Z∂M
(x)Tab(x)χa(x)nb(x)dΣx,
whe e
χa
is Killing-like o no malized bounda y ime ansla ion ec o eld,
nb
is no mal,
is posi i e weigh unc ion. Rela i e en opy
S(ω∂∥φ∂)
mono onically non-dec easing
along modula ow  u u e di ec ion, dening ime a ow on bounda y.
Pos ula e 4
(A4: Topological B anching and
Z2
Index)
.
Disc iminan subse
D⊂X
exis s in pa ame e space
X
such ha on
X◦=X D
can con inuously selec sca e ing
ma ix squa e oo
S1/2
. Any closed loop
γ⊂X◦
li ing may e u n o opposi e b anch o
o iginal poin , gi ing
Z2
index
ν(γ)∈ {±1},
his index equi alen o spec al ow pa i y o some sel -adjoin amily o in e sec ion
numbe pa i y wi h
D
.

5

4 Va ia ional Comple eness and Geome ic Bounda y
P o e: Pos ula e A1's a ia ional comple eness equi emen in oduces GHY e m and
B ownYo k bounda y s essene gy enso on non-null bounda y; when co ne s and null
bounda ies exis , need addi ional co ne and null bounda y e ms, comple ely comp ess-
ing bulk die ence in o eadou s on bounda y geome y and su ace s ess.
Theo em 4.1
(Va ia ional Comple eness on Non-Null Bounda y)
.
Unde a ia ion xing
bounda y induced me ic
hab
, o al ac ion
S[g] = SEH[g] + SGHY[g]
 s -o de a ia ion is
δS[g] = 1
16πG ZM
(Gab + Λgab)δgab√−g d4x,
i.e., all bounda y e ms comple ely cancel. Thus unde gi en
hab
condi ion, a ia ional
p inciple well-dened, de i ing Eins ein equa ion
Gab + Λgab = 8πGTab
.
B ownYo k Quasilocal S essEne gy Tenso
: A e in oducing ma e ac ion
Sma e [g, Φ]
, dene on bounda y
TBY
ab =−2
p|h|
δSGHY
δhab =1
8πG(Kab −Khab).
In eg a ing o e spa ial slice
Σ⊂∂M
gi es B ownYo k ene gy
EBY(Σ)
in e p e able
as quasilocal ene gy ela i e o e e ence backg ound, ac ing as bounda y ime ansla ion
gene a o in Hamil on o malism.
Co ne s and Null Bounda ies
: When imelike and spacelike bounda ies in e sec
o ming co ne s, o null bounda ies (like ho izons, null hype su aces) exis , GHY e m
alone insucien o ensu e a ia ional comple eness. Need in oduce co ne e m
Sco ne
and null bounda y e m
SN
.

5 Spec alSca e ing Side Scale Iden i y and Bound-
a y Time
Realize Pos ula e A2; gi e scale iden i y sucien condi ions; explain how i uni o mly
eads bounda y phase iny changes as s a e densi y die ence and  ime delay die -
ence.
5.1 Bi manK ein Spec al Shi and Sca e ing Phase
Unde Sec ion 2.2 assump ions, K ein spec al shi unc ion
ξ(λ)
dened making
( (H)− (H0)) = Z+∞
−∞
′(λ)ξ(λ)dλ
hold o sucien ly many es unc ions
. Bi manK ein o mula gi es
de S(λ) = exp(−2πi ξ(λ)).
Dene o al sca e ing phase
Φ(λ) = Pjδj(λ)
, hal -phase
φ(λ) = 1
2Φ(λ)
. Then
φ′(λ)
π=ρ el(λ).
6
5.2 Wigne Smi h Time Delay Ope a o
Recall Wigne Smi h ope a o
Q(λ) = −iS(λ)†∂λS(λ).
Taking ace yields
Q(λ)=4φ′(λ).
Combining wi h p e ious subsec ion ela ion, a anging no a ion, selec unied con-
en ion: dene ime scale densi y as
κ(ω) := φ′(ω)
π.
Theo em 5.1
(Scale Iden i y)
.
Unde s anda d assump ions ( ace-class pe u ba ion,
wa e ope a o comple eness), no maliza ion selec ion exis s making o almos all
ω
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
Physical in e p e a ion
: Scale iden i y shows bounda y phase pe u ba ion
δφ
equency change
∂ωφ
eadable by h ee equi alen ways:
•
As s a e densi y die ence
ρ el(ω)
pe uni equency;
•
As o al esidence ime  ime delay densi y
(2π)−1 Q(ω)
;
•
As ime scale densi y
κ(ω)
.
This ealizes Pos ula e A2, di ec ly in e acing bounda y sca e ing da a wi h ime
scale.

6 Modula Flow, Gene alized En opy, Bounda y Time
A ow
Explain how Pos ula e A3 o ien s ime scale on bounda y, aligning wi h sca e ing scale
iden i y.
6.1 Geome ic Exp ession o Modula Hamil onian
Le
A∂
be local ope a o algeb a associa ed wi h wedge egion o causal diamond bound-
a y,
ω∂
acuum o KMS s a e on i . In many models, modula Hamil onian
K∂
w i able
as
K∂= 2πZ∂M
ξaTabnbdΣ,
whe e
ξa
is app op ia ely no malized imelike Killing ec o o diamond-like boos ec o ,
Tab
is ene gymomen um enso . Modula ow
σω∂
(A) = eiK∂ Ae−iK∂
hus in e p e able as  he mal ime o modula ime along bounda y di ec ion.
7
6.2 Rela i e En opy Mono onici y and Time A ow
Le
ω∂, φ∂
be wo bounda y s a es; co esponding ela i e en opy
S(ω∂∥φ∂)
. In lo-
cal QFT amewo k, p o able unde es ic ion o nes ed egion amily, ela i e en opy
mono onically non-inc easing wi h egion expansion. T ansla ing o bounda y geome y,
his mono onici y go e ns gene alized en opy g ow h along ce ain  u u e di ec ions.
P oposi ion 6.1
(Modula Time A ow)
.
Assume o nes ed bounda y c oss-sec ion am-
ily
{∂M }
(e.g., sec ions ad ancing along null di ec ion)
d
d Sgen(∂M )≥0,
whe e
Sgen
is gene alized en opy; hen pa ame e
selec able as ime a ow pa ame e
on bounda y. I escaling
p opo ionally o align wi h sca e ing ime scale
κ(ω)
, can
simul aneously iew on bounda y as modula ime and sca e ing ime.

7 Topological B anching, Disc iminan Bounda y,
Z2
Index
Realize Pos ula e A4, connec ing disc iminan bounda y in pa ame e space, spec al
ow pa i y, sca e ing ma ix squa e oo b anch s uc u e.
7.1 Disc iminan Bounda y and Pa ame e Space
Conside pa ame e space
X
; each poin
x∈X
co esponds o sca e ing sys em wi h
sca e ing ma ix
S(ω;x)
. Disc iminan subse
D⊂X
exis s such ha i and only i
x∈D
,
S(ω;x)
has eigen alue
−1
nea some ene gy, degene a e eigen alues, o o he
spec al anomalies. Dene
X◦=X D
.
On
X◦
, spec al anomalies excluded; can selec p incipal b anch squa e oo  o
sca e ing ma ix
S1/2(ω;x)
sa is ying
(S1/2(ω;x))2=S(ω;x), S1/2(ω;x0)
gi en
.
7.2
Z2
Index and Spec al Flow Pa i y
Take any closed loop
γ: [0,1] →X◦
; pa allel anspo ing squa e oo
S1/2
along
γ
may
ha e o e all sign ip
S1/2(ω;γ(1)) = ±S1/2(ω;γ(0)).
Dene
ν(γ) = (+1, S1/2
no ip
,
−1, S1/2
ips
.
P oposi ion 7.1
(
Z2
Index and Spec al Flow Pa i y)
.
Unde app op ia e die en ia-
bili y and spec al gap assump ions,
ν(γ)
equals spec al ow pa i y o some sel -adjoin
amily:
ν(γ) = (−1)S ({A }),
whe e
{A }
is ela ed sel -adjoin ope a o amily cons uc ed along
γ
; spec al ow
S ({A })
eco ds numbe o eigen alues c ossing ze o.
8
Geome ic in e p e a ion
: Disc iminan
D
as pa ame e space  opological bound-
a y di ides
X◦
in o die en sec o s;
ν(γ)
eco ds whe he closed loop passes a ound
bounda y odd numbe o imes. Thus going a ound once con inuous de o ma ion com-
p essed by bounda y in o simple disc e e label
±1
.

8 Unied Theo em o Bounda y Time Geome y
Glue abo e geome ic, spec alsca e ing, modula ow, opological s uc u es in o uni-
ed bounda y ime geome y amewo k; p o e ime scale uniqueness esul .
8.1 Deni ion o Bounda y Time Geome y
Deni ion 8.1
(Bounda y Time Geome y)
.
Bounda y ime geome y consis s o da a
G∂= (∂M, hab, Kab;A∂, ω∂;S(ω); D, ν)
sa is ying:
1.
(∂M, hab, Kab)
makes g a i a ional and ma e ac ion a ia ion well-dened unde
xed
hab
condi ion, gi ing B ownYo k bounda y s essene gy enso ;
2.
(A∂, ω∂)
gi es modula ow
σω∂
and modula Hamil onian
K∂
; denes ime a ow
h ough gene alized en opy mono onici y;
3. Sca e ing ma ix
S(ω)
and spec al shi da a sa is y scale iden i y; ime scale
densi y
κ(ω)
well-dened;
4. Disc iminan
D
and
ν
dene opological b anching
Z2
index.
Call ime pa ame e
unied scale pa ame e o his bounda y ime geome y i
simul aneously scales h ee ime s uc u es:
•
G a i ygeome ic side
:
is pa ame e
along bounda y ime ansla ion ec o eld
χa
making B ownYo k ene gy change a e
unde
consis en wi h bulk ene gy ow;
•
Sca e ing side
:
ela ed o equency
ω
ia bijec ion
= (ω)
making
κ(ω)
in e p e able as
d
densi y;
•
Modula ow side
:
is modula ow pa ame e making modula Hamil onian
K∂
and B ownYo k ene gy
gene a o die only by cons an ac o .
Theo em 8.2
(Bounda y Time Scale Uniqueness Theo em)
.
Le
B= (∂M, A∂, ω∂,S∂)
be physical bounda y sys em sa is ying Pos ula es A1A4 wi h echnical assump ions:
1. B ownYo k bounda y ene gy
EBY( )
change wi h pa ame e
w i able as bounda y
ene gy ow in eg al;
2. Modula Hamil onian
K∂
compa ible wi h ime ansla ion gene a ed by
EBY
; pos-
i i e cons an
β > 0
exis s making
K∂=βEBY +
cons an
.
3. Sca e ing ma ix
S(ω)
equency dependence epa ame izable as
ω=ω( )
; scale
iden i y main ains o m unde his epa ame iza ion.
Then unique (in ane escaling sense) ime pa ame e
exis s making:
•
G a i y
geome ic side bounda y ime ansla ion;
•
Sca e ing side ime delay scale;
•
Modula
ow side modula ime ow belong o same scale equi alence class.
9