Boundary Language as a Unified Physical Framework:\\ From Scattering Phase, GHY Boundary Term to Modular Flow and Generalized Entropy as a Single Structure

Bounda y Language as a Unied Physical F amewo k:
F om Sca e ing Phase, GHY Bounda y Te m o
Modula Flow and Gene alized En opy as a Single
S uc u e
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 20, 2025
Abs ac
This pape p oposes and sys ema izes he concep o bounda y language, ew i -
ing physical heo ies as algeb aicgeome ic s uc u es abou wha is allowed o be
exchanged on causal cu su aces, wi h bulk eld heo y being me ely one ealiza-
ion o his s uc u e. Taking he bounda y obse able algeb a and bounda y s a e
as undamen al objec s, we uni y h ee seemingly independen heo e ical pa adigms
in o he same amewo k: (1) in sca e ing heo y, he spec al shi unc ion, o-
al sca e ing phase, and Wigne Smi h g oup delay; (2) in gene al ela i i y, he
GibbonsHawkingYo k (GHY) bounda y e m and B ownYo k quasilocal ene gy;
(3) in ope a o algeb as, he Tomi aTakesaki modula ow and ela i e en opy
mono onici y. The co e idea is: ime is no a pa ame e o ow wi hin he bulk ha
is gi en a p io i, bu a he a unied ansla ion pa ame e gene a ed by wha is
allowed in e ms o ux balance and in o ma ion mono onici y in he bounda y
language; all obse able delays, ene gies, and gene alized en opy a ia ions a e
die en p ojec ions o he same bounda y s uc u e.
Ma hema ically, we o malize his amewo k as h ee bounda y language ax-
ioms: (A1) Conse a ion and Flux Axiom, iewing he bounda y as a balancing
in e ace o ene gy, cha ge, and in o ma ion ux; (A2) Time Gene a ion Axiom,
iewing he one-pa ame e
∗
-au omo phism g oup dened on he bounda y and i s
gene a o as he sou ce o ime scale; (A3) Mono onici y and Consis ency Axiom,
ep esen ed by ela i e en opy mono onici y and i s geome ic o ms (quan um o-
cussing, quan um null ene gy condi ion, e c.), excluding supe causali y and nega i e
en opy anspo .
In he sca e ing ealiza ion, we p o e: he bounda y language sa is ying A1A3
necessa ily induces he scale iden i y on a well-posed sho - ange sca e ing sys em
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
whe e
φ(ω)
is he o al sca e ing hal -phase,
ρ el(ω)
is he ela i e s a e densi y,
and
Q(ω)
is he Wigne Smi h g oup delay ope a o . This iden i y unies he phase
g adien , spec al shi densi y, and g oup delay ace as a single bounda y objec
called  ime scale.
1
In he g a i y ealiza ion, we show: he GHY bounda y e m and he B own
Yo k quasilocal ene gy posi i i y a e necessa y condi ions o he bounda y language
A1A2 on he geome ic side; hus ADM ime, p ope ime, and Killing ime a e
es a ed as ansla ions gene a ed by he bounda y Hamil onian. In he ope a o
algeb a ealiza ion, we p o ide he canonical model o bounda y language h ough
Tomi aTakesaki modula heo y and implemen A3 h ough ela i e en opy mono-
onici y, he eby cha ac e izing he  ime a ow as he mono onic e olu ion o el-
a i e en opy wi h modula ime unde a na u al class o condi ions.
Finally, h ough h ee model classesone-dimensional po en ial sca e ing, s a ic
black hole ex e io egions, and Rindle wedgeswe demons a e how he bound-
a y language p oduces expe imen ally obse able ime delays, quasilocal ene gy, and
Un uh empe a u e, and gi e se e al es able spec algeome icin o ma ion he-
o e ical p edic ions. De ailed appendices p o ide p oo s o key p oposi ions such as
he sca e ingspec al shi g oup delay scale iden i y, he a ia ional comple eness
o he GHY e m, and ela i e en opy mono onici y, and in oduce he e o ge-
ome y amewo k o ni e-o de Eule Maclau in and Poisson discipline o ensu e
a con ollable mapping om bounda y eadings o expe imen al da a.
Keywo ds:
Bounda y Language; Sca e ing Phase; Wigne Smi h G oup Delay; GHY
Bounda y Te m; B ownYo k Quasilocal Ene gy; Tomi aTakesaki Modula Flow; Rela-
i e En opy; Time Scale

1 In oduc ion
1.1 Resea ch Mo i a ion and O e all S uc u e
T adi ional eld heo y and g a i a ional heo y o en s a om he bulk: gi en a man-
i old
(M, g)
wi h a me ic, a bulk ac ion
S[Φ, g]
and i s a ia ional equa ions, hen sup-
plemen ed by bounda y condi ions. Howe e , in h ee classes o seemingly un ela ed
heo ies, he bounda y has long played he ole o  uly obse able:
1. In sca e ing and spec al heo y, he o al sca e ing phase
φ(ω)
, he Bi manK en
spec al shi unc ion
ξ(ω)
, and he Wigne Smi h g oup delay ope a o
Q(ω) =
−iS†∂ωS
a e comple ely dened by he in/ou asymp o ic inni e bounda ies.
2. In gene al ela i i y, he a ia ion o he Eins einHilbe bulk ac ion equi es in o-
ducing he GibbonsHawkingYo k (GHY) bounda y e m o achie e a ia ional
comple eness, while he B ownYo k quasilocal ene gy and quasilocal s ess enso
a e s ic ly bounda y da a.
3. In ope a o algeb as and algeb aic quan um eld heo y, he Tomi aTakesaki mod-
ula ow
σω
is comple ely de e mined by he algeb as a e pai
(M, ω)
, wi h i s
pa ame e
iewed as in insic ime; he mono onici y o ela i e en opy and
gene alized en opy inequali ies depend only on he inclusion ela ionships be ween
bounda y accessible algeb as.
These ac s sugges :
he bounda y i sel , a he han he bulk, is he na u al
s age o unied physical s uc u e
. Based on his, his pape p oposes he concep
2
o bounda y language, dening physical heo y as a iple s uc u e on a ce ain causal
cu su ace
Σ⊂M
LΣ= (A∂, ω, F),
whe e
A∂
is he bounda y obse able algeb a,
ω
is he bounda y s a e, and
F
is a amily
o ux unc ionals used o cha ac e ize he exchange o ene gy, cha ge, and in o ma ion
ac oss
Σ
. Bulk eld heo y, geome y, and sca e ing cons uc ions a e me ely die en
ways o ealizing his iple.
1.2 Co e Poin : Time as Bounda y T ansla ion
The co e claim o his pape can be summa ized as:

Gi en a causal cu su ace
Σ
, physical heo y  s gi es allowed exchanges ac oss
Σ
 his is he bounda y language;

Time is no a con inuous pa ame e gi en o he bulk a p io i, bu is de i ed om
a one-pa ame e
∗
-au omo phism g oup
{α } ∈R
in e nal o he bounda y language
and i s gene a o ;

When conse a ion and mono onici y condi ions a e sa ised, his ime belongs o
he same  ime scale equi alence class as he equency de i a i e o he sca e ing
phase, he g a i a ional bounda y Hamil onian, and he modula ow pa ame e .
Mo e specically, we will p o e he scale iden i y in he sca e ing ealiza ion
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
and show in g a i y and modula ow ealiza ions: ADM/p ope ime and modula ime
can be mapped in o he same equi alence class h ough his scale.
1.3 A icle S uc u e
The s uc u e o he ull ex is as ollows: Sec ion 2 gi es he s ic deni ion o bounda y
language and he h ee axioms. Sec ion 3 ealizes he bounda y language in sho - ange
sca e ing heo y and p o es he scale iden i y. Sec ion 4 shows how he GHY e m and
B ownYo k quasilocal ene gy implemen A1A2 in g a i a ional heo y wi h bounda ies.
Sec ion 5 discusses modula ow and ela i e en opy mono onici y, demons a ing he
implemen a ion o A3 on he algeb aic side. Sec ion 6 in eg a es he h ee- e minal
s uc u e, in oducing he unied ime scale equi alence class. Sec ion 7 gi es se e al
ypical models and es able p edic ions. Appendices AD p o ide de ailed p oo s o key
p oposi ions and he e o geome y amewo k.

2 Bounda y Language and Th ee Axioms
2.1 Causal Cu Su ace and Bounda y Algeb a
Le
(M, g)
be a Lo en zian mani old wi h good causal s uc u e, and
Σ⊂M
be a
codimension-one submani old di iding space ime in o inside
Min
and ou side
Mou
.
We call
Σ
a causal cu su ace.
3
Deni ion 2.1
(Bounda y Obse able Algeb a)
.
The bounda y obse able algeb a as-
socia ed wi h he causal cu su ace
Σ
is a
C∗
-algeb a
A∂
, whose elemen s co espond o
obse ables ha can be de e mined solely h ough eadings on
Σ
, such as:

In/ou eld ope a o s and unc ions o he S-ma ix on sca e ing channels;

Bounda y-induced me ic, ex insic cu a u e, and geome ic quan i ies composed
o hem;

Local algeb as on wedge egions o Cauchy su ace bounda ies in algeb aic quan um
eld heo y.
Deni ion 2.2
(Bounda y S a e)
.
A bounda y s a e is a posi i e no malized linea
unc ional
ω:A∂→C
on
A∂
, ep esen ing he expec a ion alue o bounda y obse ables
unde gi en physical congu a ions (bulk elds, me ic, ex e nal sou ces, e c.).
2.2 Bounda y Language T iple
Deni ion 2.3
(Bounda y Language)
.
A bounda y language is he iple
LΣ= (A∂, ω, F),
whe e
A∂, ω
a e as desc ibed abo e, and
F ⊂ A∗
∂
is a amily o eal- alued linea unc ion-
als called ux unc ional amily, used o cha ac e ize bounda y eadings o exchangeable
quan i ies such as ene gy, cha ge, en opy, o in o ma ion.
Typical examples include:

In sca e ing heo y, unc ionals ela ed o p obabili y cu en , ene gy ow, o ime
delay;

In g a i a ional heo y, unc ionals ela ed o quasilocal ene gy, momen um, angula
momen um, and gene alized en opy;

In algeb aic quan um heo y, unc ionals ela ed o modula Hamil onian, ela i e
en opy, and in o ma ion ow.
2.3 Axiom A1: Conse a ion and Flux
Axiom 2.4
(Conse a ion and Flux)
.
Fo a bulk ac ion
Sbulk
and bounda y ac ion
Sbd y
sa is ying app op ia e egula i y condi ions, he e exis s a ux unc ional
F∈ F
such
ha o any compac ly suppo ed bulk a ia ion
δΦ, δg
,
δ(Sbulk +Sbd y) =
( olume in eg al)
+F(δXΣ),
whe e
δXΣ∈ A∂
is he co esponding bounda y sou ce a ia ion. I is equi ed ha o
all bounda y condi ion-sa is ying physical a ia ions, when he bulk equa ions hold,
F(δXΣ)=0
and
F
is linea o he allowed bounda y a ia ion amily, so ha bounda y a ia ion can
be in e p e ed as an exp ession o he ux conse a ion condi ion.
In ui i ely, A1 equi es ha any esidual e m o bulk a ia ion on he bounda y can
be iden ied as a ux unc ional ac ing on bounda y da a a ia ion, he eby es a ing
 a ia ional comple eness as he condi ion ha ux can be a ibu ed o bounda y
language.
4
2.4 Axiom A2: Time Gene a ion
Axiom 2.5
(Time Gene a ion)
.
On he bounda y obse able algeb a
A∂
, he e exis s a
one-pa ame e
∗
-au omo phism g oup
{α } ∈R⊂Au (A∂),
whose gene a o is a closed unbounded de i a ion
δ
, sa is ying:
1. The e exis s a amily o  ime obse ables
T ⊂ A∂
such ha o each
T∈ T
,
7→ ω(α (T))
is con inuously die en iable;
2. The ux unc ionals sa is y app op ia e in a iance o conse a ion p ope ies unde
α
, e.g., o ene gy unc ional
FE∈ F
,
FE◦α =FE;
3. The co esponding gene a o
δ
can be ep esen ed h ough a ce ain bounda y
Hamil onian
H∂∈ A′′
∂
as
d
d ω(α (A)) = i ω([H∂, α (A)])
a leas on a dense domain.
We call he pa ame e
∈R
he ime scale gene a ed by he bounda y language. I
is conjuga e o equency
ω
in he sca e ing ealiza ion, conjuga e o ADM/p ope ime
in he g a i y ealiza ion, and conjuga e o he modula pa ame e in he modula ow
ealiza ion.
2.5 Axiom A3: Mono onici y and Consis ency
Axiom 2.6
(Mono onici y and Consis ency)
.
Gi en bounda y language
LΣ
, o any wo
bounda y s a es
ω, ω′
, he e exis s a non-nega i e unc ion
S el(ω′∥ω)
called ela i e en-
opy o gene alized en opy, sa is ying:
1. Non-nega i i y:
S el(ω′∥ω)≥0
, wi h equali y i and only i
ω′=ω
;
2. Mono onici y: o any subalgeb a inclusion
A∂,1⊂ A∂,2⊂ A∂
,
S(1)
el (ω′∥ω)≤S(2)
el (ω′∥ω);
3. Time consis ency: unde app op ia e physical condi ions (such as ene gy condi ions
o KMS condi ions), i e ol ing along he ow
ω , ω′
o he ime gene a ion axiom
A2, hen
d
d S el(ω′
∥ω )≤0
a leas holds in one di ec ion (dening he ime a ow).
A3 cha ac e izes causal consis ency and in o ma ion canno inc ease as in insic
p ope ies o he bounda y language, p o iding a basis o he bounda y geome ic
algeb aic deni ion o he ime a ow.

5

3 Bounda y Language Realiza ion in Sca e ing The-
o y
3.1 Sho -Range Sca e ing and S-Ma ix
Conside sel -adjoin ope a o s
H0
and
H=H0+V
on Hilbe space
H
, whe e
V
is a
sho - ange pe u ba ion sa is ying s anda d assump ions such ha he wa e ope a o s
Ω±= s- lim
→±∞ ei He−i H0
exis and a e comple e. The S-ma ix is dened as
S= Ω∗
+Ω−,
which can be decomposed in he ene gy ep esen a ion as
S=Z⊕
S(ω) dµ(ω),
whe e
S(ω)
is a uni a y ma ix ac ing on he channel space.
He e we ake he causal cu su ace o be he asymp o ic inni e in/ou bounda y,
wi h he bounda y algeb a being
A∂=B(Hin)∨B(Hou )
o an app op ia e subalgeb a (e.g., algeb a gene a ed by asymp o ic beha io ). The
bounda y s a e
ω
can be aken as an inciden wa e packe o he mal equilib ium s a e.
Deni ion 3.1
(To al Sca e ing Phase and G oup Delay)
.
Le
Φ(ω) = a g de S(ω), φ(ω) = 1
2Φ(ω).
Dene he Wigne Smi h g oup delay ope a o as
Q(ω) = −iS(ω)†∂ωS(ω).
Deno e he ace o e he channel space as
Q(ω)
.
Also le
ξ(ω)
be he Bi manK en spec al shi unc ion, and dene he ela i e s a e
densi y as
ρ el(ω) = d
dωξ(ω)
(sign con en ion see Appendix A).
3.2 Scale Iden i y and A2 Sca e ing Realiza ion
In he sca e ing con ex , we selec  ime obse ables as obse ables gene a ed by
S(ω)
in equency space, and ux unc ionals include p obabili y cu en , ene gy ow, and
delay unc ionals.
6
Theo em 3.2
(Sca e ingSpec al Shi G oup Delay Scale Iden i y)
.
In a sca e ing
sys em sa is ying sho - ange and egula i y assump ions, he scale unc ion
κ(ω) = φ′(ω)
π
sa ises he iden i y wi h he ela i e s a e densi y
ρ el(ω)
and he g oup delay ace
(2π)−1 Q(ω)
:
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
This iden i y unies he equency de i a i e o he o al phase, he spec al shi densi y,
and he g oup delay in o he same scale unc ion
κ(ω)
, which can be iewed as he conc e e
exp ession o he ime scale gene a ed by he bounda y language a he sca e ing end.
P oo ou line
: Using he Bi manK en o mula
de S(ω) = e−2πiξ(ω),
die en ia ing wi h espec o
ω
and combining wi h
de S(ω) = eiΦ(ω)
and he deni ion
o
Q(ω) = −iS†∂ωS
, we ob ain
Q(ω)=2πξ′(ω),Φ(ω) = −2πξ(ω),
hus
φ′(ω)/π =−ξ′(ω), ρ el(ω) = −ξ′(ω),
syn hesizing o yield he s a ed iden i y. De ailed p oo in Appendix A.
Co olla y 3.3
(Sca e ing Time Scale)
.
Fo an inciden wa e packe equency-localized
in he neighbo hood o
ω0
, i s a e age g oup delay
τWS(ω0)
gi es an expe imen al eading
o he bounda y ime scale:
τWS(ω0) = Rdω|a(ω)|2 Q(ω)
Rdω|a(ω)|2≈2π κ(ω0).
3.3 Bounda y Language Pe spec i e on Spec al Flow and Topo-
logical B anches
Conside a sca e ing sys em
S(ω;λ)
ha a ies con inuously wi h pa ame e
λ∈R
.
Unde app op ia e F edholm and egula i y condi ions, he spec al shi unc ion
ξ(ω;λ)
is a con inuous unc ion o
(ω, λ)
, and i s in eg al o e
ω
gi es he spec al ow:
P oposi ion 3.4
(Spec al Flow as Bounda y Topological B anch Change)
.
O e he
pa ame e in e al
[λ1, λ2]
, he cumula i e spec al ow p oduced by eigen alue c ossings
o he spec al h eshold wi hin he ene gy in e al
[ω1, ω2]
∆N=Zω2
ω1
dω ρ el(ω;λ)
is an in ege and co esponds o a opological b anch change o he bounda y language. I s
physical meaning is: pa ame e changes cause changes in he numbe o s a es allowed o
c oss he bounda y, co esponding o ansi ions in opological classes o numbe o bound
s a es.

7
4 Bounda y Language in G a i y and GHY Te m
4.1 Va ia ional P oblem o Eins einHilbe Ac ion
Le
(M, g)
be a ou -dimensional space ime wi h bounda y
∂M
. The Eins einHilbe
bulk ac ion is
SEH[g] = 1
16πG ZM
d4x√−g R.
Fo me ic a ia ion
δgµν
, he scala cu a u e a ia ion can be w i en as
δR =gµνδRµν +Rµνδgµν,
and
δRµν
con ains  s -o de de i a i es o
δgµν
. A e in eg a ion by pa s,
δSEH
includes
olume and bounda y e ms, wi h bounda y e ms con aining
∂nδg
e ms ha canno be
exp essed solely h ough a ia ions o he induced me ic
hij =gij|∂M
, meaning
SEH
alone
canno gi e a well-dened Di ichle bounda y a ia ional p oblem.
P oposi ion 4.1
(GHY Te m as Necessa y Condi ion o A1)
.
I only
SEH
is included
wi hou adding bounda y ac ion
SGHY
, hen he bounda y con ibu ion o he me ic a i-
a ion con ains e ms ha canno be w i en as a ux unc ional
F∈ F
ac ing linea ly on
bounda y sou ce a ia ion
δhij
, hus iola ing he bounda y language axiom A1. Adding
he GHY bounda y e m
SGHY[g] = 1
8πG Z∂M
d3xp|h|K,
whe e
hab
is he bounda y-induced me ic,
K
is he ace o ex insic cu a u e,
ϵ=±1
depends on no mal ype. The bounda y a ia ion o he combined ac ion
S o =SEH +
SGHY
can be w i en as
δS o [g] =
( olume in eg al)
+1
16πG Z∂M
d3xp|h|(Kij −Khij)δhij,
whe e
Kij
is he ex insic cu a u e,
K=Kijhij
. The e o e
F(δXΣ) = 1
16πG Z∂M
d3xp|h|(Kij −Khij)δhij
cons i u es a ux unc ional, implemen ing A1. De ailed de i a ion in Appendix B.
4.2 B ownYo k Quasilocal Ene gy and Time Gene a ion
F om he a ia ion o he o al ac ion wi h espec o bounda y me ic, he B ownYo k
quasilocal ene gy-momen um enso can be dened:
TBY
ij =2
p|h|
δS o
δhij =1
8πG(Kij −Khij) +
( e e ence e m)
.
Fo a gi en ime slice
Σ ⊂∂M
and i s uni imelike ec o eld
ui
, he quasilocal
ene gy can be dened:
EBY[Σ ] = ZΣ
d2x√σ uiujTBY
ij ,
whe e
σ
is he induced wo-dimensional me ic on
Σ
.
8
Theo em 4.2
(Bounda y Hamil onian and Geome ic Time Gene a ion)
.
In space ime
wi h ADM decomposi ion, he e exis s a bounda y Hamil onian
H∂
which, unde app o-
p ia e bounda y condi ions, gene a es ime e olu ion on he bounda y algeb a h ough
Poisson b acke s o commu a o s: o any bounda y obse able
A∈ A∂
,
d
d ω (A) = i ω ([H∂, A]),
whe e
ω
is he s a e along ime slice
Σ
. Mo eo e ,
H∂
can be ob ained om he in eg al o
B ownYo k quasilocal ene gy, so geome ic ime is comple ely gene a ed by he bounda y
language, sa is ying A2.
In he s a ic case (wi h imelike Killing ec o ),
H∂
is equi alen o ADM mass o
Koma ene gy, so Killing ime, ADM ime, and he ime scale gene a ed by bounda y
language belong o he same equi alence class.

5 Ope a o Algeb as, Modula Flow, and Rela i e En-
opy
5.1 S anda d Fo m and Modula Flow
Le
M
be a on Neumann algeb a ac ing on Hilbe space
H
, and
ω
a ai h ul no mal
s a e on
M
. The GNS ep esen a ion yields ec o
Ω∈ H
such ha
ω(A) = ⟨Ω, AΩ⟩.
The Tomi a ope a o
S
is dened by
SAΩ = A∗Ω, A ∈ M
whose pola decomposi ion gi es
S=J∆1/2,
whe e
J
is conjuga ion and
∆
is he modula ope a o . The Tomi aTakesaki modula
ow is dened as
σω
(A) = ∆i A∆−i , ∈R.
In he bounda y language amewo k, we ake
A∂=M, α =σω
, ω
gi en
.
P oposi ion 5.1
(Modula Flow as Canonical Realiza ion o A2)
.
Fo any s anda d
o m
(M,H,Ω)
and ai h ul no mal s a e
ω
, he Tomi aTakesaki modula ow
{σω
}
is
a one-pa ame e
∗
-au omo phism g oup sa is ying:
1.
ω◦σω
=ω
(i.e.,
ω
is a KMS s a e o he modula ow);
2. Fo any
A∈ M
, he map
7→ ω(σω
(A))
is con inuous.
The e o e,
(M, ω, {σω
})
na u ally sa ises he ime gene a ion axiom A2, p o iding a
igo ous ma hema ical ealiza ion o modula ime.
9
C.2 Rela i e Modula Ope a o and Rela i e En opy
Gi en wo s a es
ω, ω′
, he ela i e modula ope a o
∆ω′,ω
is cons uc ed h ough he
ela i e Tomi a ope a o :
Sω′,ωAΩω=A∗Ω′,
whose pola decomposi ion is
Sω′,ω =Jω′,ω∆1/2
ω′,ω
. The A aki ela i e en opy is
S(ω′∥ω) = −⟨Ω′,log ∆ω′,ω Ω′⟩.
Rela i e en opy mono onici y can be p o en h ough comple ely posi i e ace-p ese ing
maps and S inesp ing ep esen a ion: i
Φ : M → N
is comple ely posi i e ace-
p ese ing, hen
S(ω′◦Φ∥ω◦Φ) ≤S(ω′∥ω).
Taking
Φ
as condi ional expec a ion o subalgeb a es ic ion yields Theo em 5.2.1.
C.3 Inequali y Fo m o Time A ow
In some cases, he ime a ow s a emen can be s eng hened o he ollowing inequali y:
le
ω , ω′
be s a es e ol ing along he modula ow, hen
d2
d 2S(ω′
∥ω )≥0,
i.e., ela i e en opy is con ex in modula ime. This ype o p ope y is closely ela ed
o he second-o de a ia ion o gene alized en opy in holog aphic and QNEC/QFC
li e a u e, co esponding o quan um ocussing condi ions on null bounda ies.

Appendix D: E o Geome y o Fini e-O de Eule 
Maclau in and Poisson Discipline
The scale unc ion
κ(ω)
gene a ed by bounda y language and expe imen al eadings a e
o en connec ed h ough equency in eg a ion and disc e e sampling. To ensu e igo ous
 heo yexpe imen docking, an e o geome y amewo k con olling aliasing and
unca ion e o s is needed.
D.1 Fini e-O de Eule Maclau in Fo mula
Fo sucien ly smoo h unc ions
, on in e al
[a, b]
, he Eule Maclau in o mula is
b
X
n=a
(n) = Zb
a
(x) dx+ (a) + (b)
2+
m
X
k=1
B2k
(2k)!  (2k−1)(b)− (2k−1)(a)+Rm,
whe e
B2k
a e Be noulli numbe s and
Rm
is he emainde . We en o ce aking only
ni e o de
m
and iew he uppe bound o
Rm
on a gi en unc ion class as pa o
e o geome y: i s magni ude is con olled by high-o de de i a i es o
and p incipal
singula i ies.
In p inciple, we equi e:
16

1. Any polynomial o a ional app oxima ion  ing he scale unc ion
κ(ω)
con ols
he g ow h o high-o de de i a i es wi hin physically ele an equency in e als;
2. All e o s
Rm
in oduced by Eule Maclau in unca ion do no p oduce new sin-
gula i ies, i.e., singula i y does no inc ease, poles = p incipal scales.
D.2 Poisson Summa ion and Aliasing Con ol
The Poisson summa ion o mula
X
n∈Z
(n) = X
k∈Z
ˆ
(2πk)
connec s disc e e sampling wi h equency space. Fo nume ical calcula ions o scale unc-
ions and ela ed esponse unc ions, sampling s ep
∆ω
and equency domain suppo
de e mine aliasing e o s.
In he bounda y language amewo k, we equi e:
1. Sampling sa ises he Nyquis Shannon condi ion so ha aliasing e o can be
uppe bounded;
2. Fo each expe imen al sampling scheme, gi e explici aliasing e o es ima es and
p o e hey do no in oduce new singula i ies, only changing weigh s o dis ibu-
ions;
3. E o analysis ollows ni e-o de  discipline: no elying on o mally inni e sums
o inni e die en ia ion, bu o ming closed e o geome y h ough ni e-o de
unca ion and igo ous e o bounds.
This amewo k makes he mapping om heo e ical scale unc ion
κ(ω)
o disc e e
measu emen da a ma hema ically con ollable, bo h ensu ing consis ency o bounda y
language h ee axioms in nume ical implemen a ion and p o iding di ec e o budge
ools o expe imen al design.
17