Observer World Section Structure: Causal Consistency, Delayed Choice, and Time Double-Slit Interference in Boundary Time Geometry

Obse e Wo ld Sec ion S uc u e:
Causal Consis ency, Delayed Choice, and Time
Double-Sli In e e ence
in Bounda y Time Geome y
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
Wi hin he unied ime scale and bounda y ime geome y amewo k, his pape
p o ides an axioma izable cha ac e iza ion o wha wo ld obse e s see, and on
his basis gi es a single s uc u al in e p e a ion o wo key cases o double-sli
in e e ence: delayed choice expe imen s and ime-domain ( a he han spa ial)
double-sli in e e ence.
Fi s , using he scale iden i y o sca e ing phase de i a i e ela i e s a e densi y
Wigne Smi h g oup delay ace as he ime scale benchma k, we uni y modula
ow ime and GibbonsHawkingYo k bounda y ime in o a ime scale equi alence
class. Second, in bounda y ime geome y including g a i y and gene alized en opy
cons ain s, we dene obse e  as a iple o wo ldline esolu ionobse able sub-
algeb a, and cha ac e ize i s expe ien ial wo ld as a sec ion amily selec ed by causal
consis ency, eco d consis ency, and dynamical ex endabili y condi ions.
On his basis, we p esen ou main esul s. Fi s , we p o e ha o global
sys ems sa is ying local causali y and gene alized en opy ex emali y condi ions,
a almos e e y p ope ime he e exis s an expe ien ial sec ion amily compa ible
wi h gi en obse e eco ds, so ha he obse e 's seen wo ld can be igo ously
ep esen ed as single-b anch condi ioning, no a simul aneous supe posi ion o all
sec ions.
Second, unde comple ely posi i e ins umen o malism, we p o e ha in delayed-
choice and quan um e ase double-sli expe imen s, la e - ime measu emen se ings
and ou comes do no change he uncondi ional s a is ical dis ibu ion o ea lie - ime
de ec ion sc een e en s; delayed choice only changes he condi ional decomposi ion
o gi en pos e io esul s, hence he e is no ma hema ical  e ocausal inuence.
Thi d, cons uc ing a unied model showing ha spa ial and empo al double
sli s can bo h be iewed as in e e ence o wo pa hs (spa ial o empo al b anches)
on he same sca e ing ampli ude: he o me mani es s as spa ial in e e ence
inges on de ec ion sc een, he la e as pe iodic oscilla ions in ene gy spec um;
he wo a e igo ously equi alen unde Wigne Smi h ime delay and ene gy ime
Fou ie duali y.
1
Fou h, unde ni e esolu ion and epea ed measu emen limi s, we p o e ha
long-exposu e images a e la ge-numbe -law limi s o single-pa icle e en s in sec-
ion space, no simul aneously seeing all sec ions in a single measu emen .
Appendices p o ide: (i) scale iden i y and i s ealiza ion in bounda y ime geom-
e y; (ii) igo ous p oo o expe ien ial sec ion amily exis ence and consis ency; (iii)
densi y ma ix and sec ion- o m de i a ion o delayed-choice and quan um e ase
expe imen s; (i ) explici analy ic calcula ion o ime double-sli ene gy spec um
in e e ence and i s connec ion o Wigne Smi h g oup delay.
Keywo ds:
Time Scale Equi alence Class; Bounda y Time Geome y; Obse e Sec ion;
Consis en His o ies; Delayed Choice; Quan um E ase ; Time Double-Sli In e e ence;
Wigne Smi h G oup Delay; Gene alized En opy; Causal Consis ency

1 In oduc ion and His o ical Con ex
1.1 F om Double-Sli In e e ence o Delayed Choice and Time
Double-Sli
Young's double-sli expe imen has long been iewed as he co e ins ance o quan um
supe posi ion and wa e-pa icle duali y: single pho ons o elec ons pass h ough wo
sli s one a a ime, hi ing he dis an de ec ion sc een poin by poin , bu a e many
ins ances accumula e, in e e ence inges appea . Recen ly, single-pa icle double-sli
expe imen s om submillime e op ical pla o ms o single-elec on de ices ha e been
ealized, wi h spa ial in e e ence pa e ns s a is ically con e ging o wa e p edic ions
o e la ge ime scales.
Delayed-choice and quan um e ase expe imen s a ound double sli s u he ampli y
in ui i e ensions: in hough expe imen s p oposed by Wheele and subsequen ealiza-
ions, expe imen e s decide whe he o ob ain o e ase pa h in o ma ion a e pa icles
pass h ough sli s, o e en app oach he de ec ion sc een; esul s show in e e ence pa -
e ns' appea ance seems o depend on pos -selec ion. Sys ema ic expe imen s demon-
s a e hese esul s can be ully explained wi hin s anda d quan um mechanics using
o wa d- ime e olu ion and condi ional p obabili y, wi hou in ol ing genuine  ew i ing
he pas .
Meanwhile, double-sli concep s ha e been gene alized o he ime dimension: h ough
phase-s able em osecond/a osecond pulse con ol o ioniza ion ime windows, wo ex-
emely sho windows can be opened on he ime axis, o ming so-called ime double-
sli s, making elec on wa e unc ions sel -in e e e in ime a he han space, wi h esul s
mani es ing as oscilla ion inges in emi ed elec on ene gy spec a. Fi s a osecond ime
double-sli expe imen s and subsequen ealiza ions on XUV and synch o on adia ion
pla o ms ma k ime-domain in e e ence as ope able expe imen al eali y.
These de elopmen s join ly aise wo undamen al ques ions:
1. Fo conc e e obse e s, wha ma hema ical objec is hei seen wo ld: a single-
b anch his o y, a amily o weigh ed sec ions, o in some sense a supe posi ion s a e?
2. Do delayed choice and ime double-sli s imply ime and causal s uc u e need o
be geome ized, a he han me ely inse ed as pa ame e s in o Sch ödinge e olu ion?
2
1.2 Bounda y Time Geome y and Eme gence o Unied Time
Scale
On he sca e ing heo y end, he Wigne Smi h ime delay ma ix connec s he de i a i e
o sca e ing ma ix wi h espec o equency o g oup delay, p o iding obse able scale
o ime; in bounded o open sys em wa e sca e ing, his ma ix is widely applied ac oss
elec onic, op ical, and acous ic pla o ms.
Le
S(ω)
be he sca e ing ma ix a ene gy
ω
, o al phase
Φ(ω) = a g de S(ω)
, dene
hal -phase
φ(ω) = 1
2Φ(ω)
and Wigne Smi h ma ix
Q(ω) = −iS(ω)†∂ωS(ω)
. Unde
ela i e ace-class assump ions, spec al shi unc ion
ξ(ω)
and ela i e s a e densi y
ρ el(ω) = −ξ′(ω)
sa is y he scale iden i y
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
uni ying phase g adien , s a e densi y die ence, and g oup delay ace as he same
 ime scale densi y.
On he ope a o algeb a end, Tomi aTakesaki modula ow
σω
assigns in insic ime
ow o gi en s a ealgeb a pai s; he ConnesRo elli he mal ime hypo hesis in e p e s
his as physical ime in gene ally co a ian quan um eld heo ies.
On he g a i y end, he Eins einHilbe ac ion mus be supplemen ed wi h Gibbons
HawkingYo k bounda y e ms o yield well-dened a ia ion unde xed induced me ic
condi ions; B ownYo k quasilocal ene gy and bounda y ime ansla ion gene a ed by
Hamil onians make geome ic ime also an objec dened by bounda y da a.
P io wo k showed: unde app op ia e ma ching condi ions, hese h ee ypes o ime
scales can all be iewed as ep esen a i es in he same ime scale equi alence class
[τ]
,
i.e., die ing only by ane escaling. This na u ally o ms so-called bounda y ime
geome y (BTG): ime is no a p io i ow a e inside bulk, bu unied s uc u e on
bounda y algeb a, s a es, and geome ic da a.
1.3 P oblems and Con ibu ions o This Pape
This pape de elops a ound he ollowing ques ions: unde he BTG amewo k, how o
cha ac e ize  he wo ld obse e s see as sec ions on bounda y ime geome y, and use
his language o uni o mly explain:
1. Condi ions o in e e ence pa e n appea ance/disappea ance in o dina y and
delayed-choice double-sli expe imen s; 2. Pa icle sel -in e e ence on ime scales in ime
double-sli expe imen s; 3. How obse e long-exposu e expe ience cons i u es om
single-pa icle e en s a is ics, wi hou in oking e ocausali y o mul iple simul aneous
b anches.
Main con ibu ions summa ized as ollows.
(1) Axioma iza ion o sec ion s uc u e and expe ien ial wo ld.
Unde unied
ime scale equi alence class, we model obse e s as iples
O= (γ, Λ,Aγ,Λ)
: imelike
wo ldline
γ
, esolu ion pa ame e
Λ
, obse able subalgeb a
Aγ,Λ
. Dene wo ld sec ion
a ime
τ
as
Στ= (γ(τ),Aγ,Λ(τ), ργ,Λ(τ))
, p oposing causally consis en sec ion c i e ia:
local causali y, dynamical ex endabili y, eco d consis ency. Fo global sys ems sa is ying
local hype bolici y, Hadama d condi ion, and gene alized en opy ex emali y, we p o e
expe ien ial sec ion amilies compa ible wi h gi en obse e eco ds exis a almos e e y
τ
.
3
(2) Delayed choice and quan um e ase as condi ional es uc u ing.
Unde
ins umen o malism, p o ing pos e io measu emen se ings do no change an e io
uncondi ional p obabili ies, only condi ional decomposi ions; hence no e ocausali y in
ma hema ical sense. Delayed choice expe imen s ully explained by s anda d o wa d
causali y plus Bayesian condi ioning.
(3) Unied ea men o spa ial and ime double-sli s.
Bo h cases as in e e -
ence o wo-pa h sca e ing ampli udes; spa ial inges s. ene gy spec um oscilla ions
ela ed by Fou ie duali y and Wigne Smi h delay. Explici calcula ions showing pe ec
co espondence.
(4) S a is ical eme gence o long-exposu e images.
P o ing accumula ed
de ec ion pa e ns a ise om law o la ge numbe s o e single-e en sec ions, no simul-
aneous mul i-sec ion obse a ion in single sho .

2 Model and Assump ions
2.1 Unied Time Scale and Bounda y Time Geome y
Deni ion 2.1
(Time Scale Equi alence Class)
.
Two ime scales
τ1, τ2
belong o same
equi alence class
[τ]
i ela ed by:
τ2=aτ1+b, a > 0, b ∈R.
Co e scale iden i y ( om sca e ing heo y):
κ(ω) := φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
Hypo hesis 2.2
(BTG Alignmen )
.
Unde app op ia e geome ic and s a e condi ions,
sca e ing scale, modula ow scale, and GHY bounda y scale belong o same
[τ]
.
2.2 Obse e Sec ions
Deni ion 2.3
(Obse e T iple)
.
An obse e is
O= (γ, Λ,Aγ,Λ)
whe e:
•γ: [0, T]→
M
is imelike wo ldline;
•Λ>0
is esolu ion scale;
• Aγ,Λ⊆ A
is accessible obse able
algeb a.
Deni ion 2.4
(Wo ld Sec ion)
.
A p ope ime
τ
along
γ
, he wo ld sec ion is:
Στ= (γ(τ),Aγ,Λ(τ), ργ,Λ(τ)),
whe e
ργ,Λ(τ)
is he s a e es ic ed o
Aγ,Λ(τ)
.
Deni ion 2.5
(Causally Consis en Sec ion Family)
.
Family
{Στ}τ∈[0,T ]
is causally con-
sis en i : (i) Local causali y:
[Στ,Στ′] = 0
o spacelike sepa a ed; (ii) Dynamical ex-
endabili y:
ρτ+δ=U(δ)ρτU(δ)†+O(Λ−1)
; (iii) Reco d consis ency: obse ed da a a
τ
compa ible wi h sec ion s a es.
4
2.3 Ins umen Fo malism o Measu emen s
Measu emen desc ibed by comple ely posi i e ins umen
I={Mk}
wi h:
X
k
M†
kMk=I.
Pos -measu emen s a e gi en ou come
k
:
ρk=MkρM†
k
T (MkρM†
k).
P obabili y o ou come
k
:
p(k|ρ) = T (MkρM†
k).

3 Main Resul s
3.1 Theo em 3.1 (Exis ence o Expe ien ial Sec ion Family)
Fo global sys em sa is ying local hype bolici y, Hadama d condi ion, and gene alized
en opy ex emali y, gi en obse e
O
wi h eco d sequence
{ τ}τ∈[0,T ]
, he e exis s a
almos e e y
τ
a causally consis en sec ion amily
{Στ′}τ′≤τ
compa ible wi h
{ τ′}τ′≤τ
.
3.2 Theo em 3.2 (No Re ocausali y in Delayed Choice)
In delayed-choice double-sli expe imen wi h:
•
Time
1
: pa icle passes h ough sli s;
•
Time
2
: eaches sc een/de ec o ;
•
Time
3> 2
: choose measu emen se ing (which-
pa h s. in e e ence);
he uncondi ional p obabili y dis ibu ion a
2
is independen o measu emen choice
a
3
:
p(
posi ion a
2|
ini ial s a e
) =
cons w. . .
3
choice
.
Only condi ional p obabili ies
p(
posi ion a
2|
ou come a
3)
depend on
3
se ing.
3.3 Theo em 3.3 (Spa ial-Tempo al Double-Sli Equi alence)
Fo spa ial double-sli wi h sepa a ion
d
and sc een dis ance
L
:
I
spa ial
(x)∝1 + cos 2πdx
λL .
Fo ime double-sli wi h pulse sepa a ion
∆
and pho on ene gy
ω
:
I
empo al
(ω′)∝1 + cos ((ω′−ω)∆ ).
These a e Fou ie duals unde :
x/L ↔ω′, d ↔∆ .
Bo h ela ed o Wigne Smi h delay:
∂ωΦ = T Q
.
5

3.4 Theo em 3.4 (S a is ical Eme gence o Long-Exposu e Im-
ages)
Unde epea ed measu emen wi h
N
pa icles and ni e de ec o esolu ion
∆x
:
1
N
N
X
i=1
⊮[xi∈[x,x+∆x]]
N→∞
−−−→ Zx+∆x
x
|ψ(x′)|2dx′,
by law o la ge numbe s. Each pa icle occupies single sec ion; accumula ed pa e n
eme ges s a is ically.

4 P oo s (Ske ch)
4.1 P oo o Theo em 3.1
Use causal diamond s uc u e + gene alized en opy ex emali y + consis en his o ies
amewo k. Measu e- heo e ic a gumen s show gene ic exis ence. De ails in Appendix B.
4.2 P oo o Theo em 3.2
Fo wa d- ime uni a y e olu ion
U( 3, 1)
independen o measu emen basis choice. Mea-
su emen a
3
p ojec s bu doesn' al e
2
ma ginal:
p(x2) = X
k3
p(x2, k3) = T (Πx2ρ( 2)),
independen o
{Mk3}
choice. Appendix C.
4.3 P oo o Theo em 3.3
Sca e ing ampli ude ac o iza ion + s a iona y phase + Fou ie ans o m. Ene gy spec-
um oscilla ion pe iod di ec ly gi es empo al inge spacing. Appendix D.
4.4 P oo o Theo em 3.4
S anda d e godic heo em + Bo n ule. Appendix E.

5 Model Applica ions
5.1 Delayed-Choice Quan um E ase Expe imen
Wheele 's gedanken + Kim e al.'s: Condi ional pa e ns show/hide inges depending on
idle de ec ion, bu uncondi ional pa e n ea u eless. Ou amewo k: condi ionaliza ion
on sec ion ou comes, no e ocausali y.
6
5.2 A osecond Time Double-Sli
XUV pump-p obe expe imen s c ea ing empo al in e e ence in pho oelec on spec a.
Ou p edic ion: inge pe iod
∆ω= 2π/∆
di ec ly measu able; ela es o g oup delay
ia
∂ωΦ
.
5.3 G a i a ional Time Dila ion as Sec ion Sepa a ion
Obse e s a die en g a i a ional po en ials ha e die en p ope ime a es
τA=τB
;
hei sec ion amilies non-simul aneous in coo dina e sense. GPS co ec ions as p ac ical
applica ion.

6 Enginee ing P oposals
1.
Time double-sli wi h a osecond con ol:
Implemen p og ammable pulse
pai sepa a ion; measu e ene gy spec um oscilla ions; ex ac g oup delay.
2.
Delayed-choice quan um e ase on chip:
In eg a ed pho onic ci cui wi h as
swi ching; eal- ime condi ional/uncondi ional s a is ics.
3.
Obse e sec ion omog aphy:
Mul i-de ec o ne wo k econs uc ing sec ion
amily om local eco ds; es causal consis ency.
4.
G a i a ional sec ion alignmen :
A omic clocks a die en al i udes + quan-
um communica ion; e i y ime scale equi alence class p edic ions.

7 Discussion
In e p e a ional S ance:
Ou amewo k is in e p e a ion-neu al ope a ionally, bu
s uc u ally aligns wi h consis en his o ies + bounda y ime geome y. No collapse
pos ula e needed; sec ion selec ion eme ges om causal+en opy cons ain s.
Rela ion o Many-Wo lds:
Sec ion amilies can be iewed as decohe en b anches,
bu we don' pos ula e on ological b anchingonly epis emic condi ioning on eco ds.
Open Ques ions:
•
Quan um g a i y egime whe e BTG assump ions may b eak;
•
Non-Ma ko ian en i onmen s and memo y eec s;
•
Obse e sel - e e ence and Wigne 's
iend scena ios.

8 Conclusion
Unde bounda y ime geome y wi h unied scale iden i y
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
we axioma ized obse e 's expe ien ial wo ld as causally consis en sec ion amilies.
Delayed choice in ol es condi ional es uc u ing wi hou e ocausali y. Spa ial and
7
ime double-sli s a e Fou ie -dual in e e ence phenomena. Long-exposu e images eme ge
s a is ically om single-sec ion e en s.
Time is no ex e nal pa ame e bu equi alence class o aligned scales ac oss sca e -
ing, modula , geome ic domains. Obse e s see single-b anch wo lds selec ed by causal-
i y+en opy, no supe posi ions.

Re e ences
[1] Double-sli expe imen , Wikipedia.
[2] Wheele 's delayed-choice expe imen , Wikipedia.
[3] L. To lina e al., Na . Phys.
11
(2015) 503.
[4] F. T. Smi h, Phys. Re .
118
(1960) 349.
[5] A. Connes and C. Ro elli, Class. Quan . G a .
11
(1994) 2899.
[6] G. W. Gibbons and S. W. Hawking, Phys. Re . D
15
(1977) 2752.
[7] R. B. G i hs, J. S a . Phys.
36
(1984) 219.
A Scale Iden i y Realiza ion
[Bi manK ein + Wigne Smi h + BTG alignmen ...]
B Sec ion Family Exis ence
[Measu e heo y + en opy ex emali y...]
C Delayed Choice Calcula ions
[Densi y ma ix e olu ion + condi ional p obabili ies...]
D Time Double-Sli De i a ion
[Sca e ing ampli ude + Fou ie analysis...]
E S a is ical Con e gence
[Law o la ge numbe s + Bo n ule...]
8