Representation and Inversion of ``Time'' in EBOC

Rep esen a ion and In e sion o “Time” in EBOC
— Cha ac e izing “Sequence” and “Choice” in a S a ic Block Uni e se,
and De i ing Consciousness Sel -Linea iza ion om Recu si e Un olding
o Obse a ion Windows
Anonymous Au ho
Ve sion: 2.7
Abs ac
In a “ imeless” s a ic block uni e se (EBOC), all ac s a e gi en as once- o -all s uc u e–
measu e objec s; so-called “ ime” should be a seconda y calib a ion endogenously in e ible
om ha objec , no a p imi i e coo dina e. This pape p o ides a igo ous ou e unde
he uni ied seman ics o EBOC: i s , ia he window–consensus pa adigm, we de ine
“sequence” as a bidi ec ionally in ini e pa h (consensus chain) on he unc ion g aph
d i en by a uni ied selec o , ensu ing “unique successo ” h ough p e e ence agg ega ion
and well-o de disambigua ion; hen, ia he iden i y windowed ace = phase–densi y
calib a ion (phase de i a i e = ela i e densi y o s a es = Wigne –Smi h g oup delay
ace), we es ablish “ ime eadou ” as window-weigh densi y in eg al and close i un-
de ini e-o de Nyquis –Poisson–Eule –Maclau in e o discipline; inally, in KL/B egman
in o ma ion geome y, we cha ac e ize he ecu si e un olding o obse a ion windows
as an I-p ojec ion (minimal KL) sequence, he eby ob aining he consciousness sel -
linea iza ion heo em and in e sion pa ame e s in dual (expec a ion) coo dina es.
The co e conclusion is: he one-dimensionali y o “na a i e ime” in EBOC can be endoge-
nously in e ed ia he combined o ce o “s uc u al selec ion + me ic eadou ”.
1 No a ion & Axioms / Con en ions
(Calib a ion Ca d I: T ini y) The calib a ion iden i y holding almos e e ywhe e in he
absolu ely con inuous spec um
φ′(E)
π=ρ el(E) = 1
2π Q(E),Q(E) = −i S(E)†dS
dE (E).
whe e S(E) is he sca e ing ma ix, φ′(E) is he o al sca e ing phase de i a i e, ρ el is he
ela i e densi y o s a es; he iden i y a ises on one hand om he Bi man–K e˘ın o mula
de S(E) = e2πiξ(E)⇒ξ′(E) = ρ el(E) = 1
2πi
d
dE log de S(E).
De ine o al sca e ing phase φ(E) := 1
2ilog de S(E) (con inuous b anch), hen
φ′(E)
π=1
2πi
d
dE log de S(E) = ξ′(E) = ρ el(E),
ully consis en . On he o he hand om Wigne –Smi h ime-delay ma ix and K e˘ın–F iedel
ela ion ρ el(E) = 1
2π Q(E) [?].
1
(Calib a ion Ca d II: NPE Fini e-O de Discipline) All windowed compu a ions only
allow ini e-o de Eule –Maclau in (EM) and Poisson summa ion; e o s ic ly decomposes
as
ε=εalias +εEM +ε ail,
whe e unde Nyquis sampling (band-limi ed signal, sampling a e >2B)εalias = 0; EM emain-
de con olled by Be noulli polynomials and highe -o de de i a i es o in eg and; ail con olled
by as decay and band limi a ion. This discipline gua an ees non-inc easing singula i y and
“pole = p ima y scale” [?].
Windows and ke nels. On ene gy axis RE, gi en e en window wR≥0 and on -end
ke nel h≥0 (band-limi ed, egula , and RRh(E)dE = 1), con olu ion deno ed (h⋆ρ)(E).
Wo king ene gy band. Deno e
B:= ess suppXkwRk⊂RE,
he essen ial suppo o poin wise weigh sum o he window amily. All asse ions in his pape
abou co e age,bounded o e lap (s ong/weak), eadou and in e sion a e s a ed on B.
In eg abili y. Assume ρ el ∈L1
loc(B); acco dingly all RE( )
E0ρ el appea ing in his pape a e
well-de ined on B.
Window amily co e age. Le window amily {wRk}sa is y X
k
wRk(E)>0 a.e. on B.
Window amily bounded o e lap. S ong o m: he e exis s C < ∞such ha
X
k
wRk(E)≤Ca.e.; Weak o m: he e exis M < ∞and Wmax <∞such ha o
any E, #{k:wRk(E)>0} ≤ Mand supk∥wRk∥∞≤Wmax. Unde ei he condi ion and
h≥0,Rh= 1, we ha e X
k
wRk(E)h⋆ρ el(E)∈L1
loc;acco dingly, Fde ined in
§
?? is
a locally bounded a ia ion (absolu ely con inuous) unc ion on B; i u he assum-
ing ZBX
k
wRk(E)h ⋆ ρ el(E)
dE < ∞(e.g., ini e window amily, o PkwRk∈L1(B) and
ρ el ∈L1(B)), hen Fis globally bounded a ia ion on B.
Window amily no maliza ion (PUC) and app oxima e iden i y ke nel. Le
PkwRk(E)≡1 a.e. on B, ake nonnega i e ke nel amily {hε}ε>0sa is ying Rhε= 1 and
o all ∈L1
loc(B) we ha e hε⋆ → in L1
loc(B) (ε→0). Unde PUC and NPE ini e-o de
discipline, he band-limi ed quan i y hε⋆ ρ el sa is ies
Fε(E) := X
kZE
−∞
wRk(E′)hε⋆ ρ el(E′)dE′=ZE
−∞
ρ el(E′)dE′+Cε+O(εEM +ε ail),
whe e cons an Cε oge he wi h EM/ ail e ms gi e a uni o m uppe bound, and Cε→C0as
ε→0.
F ames and band limi a ion. Mul i-window Gabo / ame Pa se al/Tigh cons uc ion
and Wexle –Raz bio hogonali y p o ide s abili y and densi y c i e ia o windowed econs uc-
ion and mul i-channel coope a ion; c i ical sampling cons ained by Balian–Low phenomenon
[?].
In o ma ion geome y. Adop Legend e po en ial Λ and B egman/KL cons uc ion: ∇Λ
gi es expec a ion coo dina es, I-p ojec ion is minimal KL unde linea momen cons ain s;
KKT condi ions cha ac e ize unique op imal poin and gi e sensi i i y [?].
2
2 Timeless Cha ac e iza ion o “Sequence” and “Choice”
2.1 Window G aph, Causal Compa ibili y, and Feasible Pa hs
Take window adius and allowed agmen se C. Cons uc De B uijn- ype window g aph
Γ: e ices a e local agmen s o leng h 2 , edges a e one-s ep slides; impose causal compa i-
bili y (ad ancing along edges does no iola e unde lying dependency p eo de ). Thus easible
sequences X:Z→ C on he block co espond one- o-one wi h bidi ec ional pa hs on Γ [?].
2.2 Uni ied Selec o and Func ion G aph Decomposi ion
Fo each e ex, agg ega e mul i-agen p e e ences as weigh ed ex emum, and disambigua e
wi h well-o de , ob aining uni ied selec o Sel and de e minis ic successo ; his yields
unc ion g aph ΓSel (each poin ou -deg ee = 1). Any ini e ou -deg ee-1 di ec ed g aph de-
composes in o se e al di ec ed cycles plus hei in- ees; pe iodic poin s o m cycles, o he s
a e ansien nodes. This pape de ines bidi ec ionally in ini e consensus chain as bidi ec-
ionally ex ended pa hs on cycles [?].
P oposi ion 2.1 (Func ion G aph S uc u e–Fini e F agmen Case).Le allowed agmen
se Cbe ini e (equi alen ly: alphabe ini e and window adius ini e), hen each connec ed
componen o ΓSel con ains exac ly one di ec ed cycle, wi h o he e ices lowing in o ha
cycle ia di ec ed ees; he cycle admi s bidi ec ional in ini e pa hs, called consensus chains.
Gene al in ini e case: each connec ed componen con ains a mos one di ec ed cycle.
P oo . Func ion g aphs a e s anda d unc ional dig aphs; hei decomposi ion p ope ies a e as
s a ed in he li e a u e (cycles + in- ees) [?].
2.3 Linea Ex ension and Th eshold S abili y
Fo dependency p eo de ⪯, by Szpil ajn’s heo em any pa ial o de ex ends o a o al o de ; on
he consensus chain image se ake his consis en linea ex ension as he index coo dina e
∈Z. When weigh s and disambigua ion ha e minimal gap, unique successo emains in a ian
unde small pe u ba ions ( h eshold s able) [?].
De ini ion 2.2 (Sequence and Choice).Choice: Gi en window s a e ,Sel( )selec s unique
successo edge;
Sequence: Bidi ec ional pa h ( ) ∈Zon ΓSel sa is ying → +1.
3 Rep esen a ion o “Time”: Phase–Densi y–Windowed T ace
3.1 Phase De i a i e = Rela i e Densi y o S a es = G oup Delay T ace
On he absolu ely con inuous spec um, he Bi man–K e˘ın o mula connec s spec al shi unc-
ion ξand S(E):
de S(E) = e2πiξ(E)⇒ξ′(E) = ρ el(E) = 1
2πi
d
dE log de S(E).
On he o he hand, Wigne –Smi h de ines Q(E) = −iS†S′, K e˘ın–F iedel ela ion gi es ρ el(E) =
1
2π Q(E). Toge he yield he iden i y in Calib a ion Ca d I [?].
3
3.2 Windowed Readou and Non-Asymp o ic Closu e
De ine windowed ace eadou
Obs(R;ρ el) := ZR
wR(E)h⋆ρ el(E)dE.
Disc e e implemen a ion obeys NPE h ee-way decomposi ion: aliasing e m (Poisson side),
bounda y Be noulli laye (EM side), and ail (band-limi ed decay). When sampling sa is ies
Nyquis , εalias = 0; EM emainde con olled by Be noulli coe icien s and highe -o de de i a-
i e bounds; ail con olled by band limi a ion and window egula i y, hus no new singula i ies
in oduced [?].
De ini ion 3.1 (EBOC Time Readou Func ional).Gi en window amily {wRk}and ke nel h,
de ine
T[ρ el] := X
kZR
wRk(E)h⋆ρ el(E)dE,
and unde addi ional in eg abili y assump ion ZB
|h⋆ρ el|dE < ∞,X
k
wRk∈L∞(B)∩L1(B)
(o ini e window amily), T[ρ el]is ini e and can be gi en a uni o m uppe bound ia NPE
ini e-o de e o ; unde Nyquis εalias = 0 [?].
4 In e sion o “Time”: Reco e ing Linea O de om Window
Da a
4.1 Phase In eg al Index
Le consensus chain C={ } ∈Z. In B,E( ) is aken om he mono one p eimage o he
eadou unc ional: deno e
F(E) := X
kZE
−∞
wRk(E′)h⋆ρ el(E′)dE′.
Unde band limi a ion, Nyquis sampling, wRk≥0, h ≥0and window amily bounded
o e lap (s ong o weak o m),Fis locally bounded a ia ion (absolu ely con inu-
ous) on B; i u he sa is ying he global in eg abili y condi ion gi en in he p e ious sec ion,
hen Fis globally bounded a ia ion on B. On he absolu ely con inuous pa o B,
ρ el =ξ′a.e. holds, hence ρ el ≥0a.e. ⇔ξis non-dec easing he e. Unde his p emise
Fis mono one non-dec easing; i u he adding window amily co e age and ρ el >0
(a.e.), hen Fis s ic ly mono one. Selec s ep calib a ion ∆ >0, de ine e ec i e index
se
TF:= { ∈Z| ∆∈ anF|B}.
Fo ∈ TF, ake igh -con inuous gene alized in e se
E( ) := F−1( ∆), F −1(y) := in {E:F(E)≥y}.
To elimina e addi i e cons an , ake baseline index ⋆∈ TFand se E0:= E( ⋆), acco dingly
de ine
τ( ) := ZE( )
E0
ρ el(E)dE =1
2πZE( )
E0
Q(E)dE,
by ρ el ∈L1
loc(B) in No a ion, he abo e in eg al is well-de ined on B. Thus τ( ⋆) = 0; i
ρ el ≥0(a.e.) hen τis mono one non-dec easing, and when window amily co e age
and ρ el >0 (a.e.) hold, τis s ic ly inc easing [?].
4
Unde he gene al condi ion o only sa is ying “window amily co e age + bounded o e lap”,
Fp o ides s ic ly mono one ene gy pa ame e E( ) and o de equi alence; phase coo dina e
τs ill needs o be cons uc ed h ough Rρ el. I u he sa is ying PUC + app oxima e
iden i y ke nel, hen he e exis s cons an Csuch ha
τ( ) = F(E( )) −F(E0) + O(εEM +ε ail),
hus “ ime eadou ” can be di ec ly gi en by p e ix windowed eadou (up o cons an ) wi h
e o uni o mly con olled by NPE discipline.
4.2 In e sion Theo em
Theo em 4.1 (Time In e sion).Unde band-limi ed windows, Nyquis sampling, and ini e-
o de EM condi ions:
(1) (Gene al condi ion) The linea o de o any consensus chain Ccan be in e ed om he
p e ix windowed eadou F ia he gene alized in e se F−1 o a s ic ly mono one ene gy
pa ame e E( ), and acco dingly ob ain a bounded a ia ion pa ame e equi alen o he
chain index ; i ρ el ≥0(a.e.), his pa ame e is mono one non-dec easing, and when
window amily co e age and ρ el >0(a.e.) hold, i is s ic ly inc easing.
(2) (Addi ional PUC + app oxima e iden i y ke nel) Fu he we ha e τ( ) = F(E( )) −
F(E0) + O(εEM +ε ail), hus phase coo dina es can be di ec ly eco e ed om F(o i s
Nyquis sampling {F(Ej)}) wi hin a uni o m e o bound.
P oo ske ch. (i) By Calib a ion Ca d I and K e˘ın–F iedel ela ion, educe windowed ace o
Rρ el; (ii) Nyquis closes aliasing o ze o, EM emainde and ail con olled; (iii) By in eg abili y
assump ion ρ el ∈L1
loc(B) we know τis well-de ined on B; when ρ el ≥0(a.e.),τis mono one
non-dec easing; unde window amily co e age and ρ el >0 (a.e.), τis s ic ly mono one
and in e ible [?].
5 Recu si e Un olding o Obse a ion Windows ⇒Conscious-
ness Sel -Linea iza ion
5.1 Submission = I-P ojec ion (Minimal KL)
Le in e nal s a e be ep esen ed by na u al pa ame e θ, po en ial unc ion Λ o Legend e ype,
expec a ion coo dina e X=∇Λ(θ). Each obse a ion s ep upda es a ge momen Fn o Fn+1,
submission/collapse equi alen o
θn+1 = a g min
θKL(Pθ∥Pθn) s. . Eθ[T] = Fn+1,
i.e., I-p ojec ion on linea cons ain s; unique solu ion exis s and sa is ies KKT. B egman–
Euclidean Py hago ean p ope y gi es op imal decomposi ion o p ojec ion [?].
5.2 Linea Response and Quasi-Linea T ajec o y
When window change is “mild” and NPE o de ixed, hen
Xn+1 −Xn=∇2Λ(θn) (θn+1 −θn) + o∥θn+1 −θn∥,
KKT and s ong con exi y gi e i s -o de linea esponse; epa ame izing he i e a ion wi h
τ om
§
??, can be iewed as app oxima e equal-s ep ad ance along some ixed ec o ∗in
expec a ion coo dina es.
5

Theo em 5.1 (Consciousness Sel -Linea iza ion).Le Λbe essen ially smoo h s ic ly con ex
po en ial; windows wRand ke nel hband-limi ed and sa is y Wexle –Raz/Pa se al ame s abil-
i y; sampling Nyquis . Then he submission s a es {Xn}d i en by ecu si e windows admi a
s ic ly inc easing epa ame iza ion map σ:Z→Zand cons an ec o ∗in expec a ion
coo dina es, and he e exis s unc ion εmic o(R, ∆) = oR→∞,∆→0(1), such ha o any baseline
nand all in ege s m
∥Xσ(n+m)−Xσ(n)−m ∗∥ ≤ |m|hCεEM +ε ail+εmic o(R, ∆)i.
Con en ion: εmic o(R, ∆) depends only on window/ke nel and NPE o de , sa is ying εmic o(R, ∆) →
0(as R→ ∞,∆→0wi h NPE o de ixed), he bound e lec s linea accumula ion o e o
wi h s ep coun |m|. Acco dingly, consciousness exhibi s quasi-linea dominan ajec o y
in i s own dual coo dina es.
P oo ske ch. Wexle –Raz and Pa se al/Tigh gua an ee eadou mapping and econs uc ion
s abili y; KKT and B egman geome y’s Py hago ean iden i y gi e i s -o de linea iza ion o
each I-p ojec ion s ep; NPE cons ain ensu es noise e ms a e en i ely domina ed by ini e-o de
emainde [?].
6 Uni ied T ini y o “Sequence–Choice–Time”

F om block o sequence: Uni ied selec o gene a es consensus chain (bidi ec ionally
in ini e pa h) on unc ion g aph, and ia Szpil ajn assigns consis en linea ex ension;

F om sequence o ime: Phase–densi y–g oup delay calib a ion iden i y makes “ ime
eadou ” become in eg al o window-weigh ed densi y; Nyquis –EM gua an ee non-asymp o ic
closu e;

F om ime o consciousness: I-p ojec ion on ecu si e windows enables dual coo di-
na es o acqui e quasi-linea p incipal axis, wi h τas he endogenous pa ame e o ha
axis.
7 Th esholds, Singula i ies, and Implemen a ion No es
1. Th eshold/ esonance: Singula i ies (such as poles, b anch poin s) o φ′(E) (equi -
alen ly ρ el(E)) co espond o con inuous spec um h esholds and esonances; ze os do
no cons i u e gene al c i e ia. Windowing and ini e-o de EM do no inc ease singula i y,
main aining “pole = p ima y scale” [?].
2. F ames and densi y: Mul i-window Pa se al/Tigh and Wexle –Raz bio hogonali y
gua an ee obus econs uc ion; c i ical sampling cons ained by Balian–Low, edundan
sampling in e als ecommended [?].
3. Sampling and aliasing: Band limi a ion and Nyquis sampling a e su icien condi ions
o closing aliasing; in enginee ing implemen a ion, modula ion–downsampling s a egy
can achie e in-band Nyquis [?].
8 Conclusion
F om he EBOC s a ic block pe spec i e, “ ime” is no a p imi i e axis bu gene a ed in h ee
s eps: (i) window–consensus condenses choice in o unc ion g aph’s consensus chain;(ii)
phase–densi y enables he chain o acqui e in e ible ime calib a ion (windowed ace ead-
ou ); (iii) KL/B egman makes submission o ecu si e windows exhibi sel -linea iza ion
6
in dual coo dina es. This ou e is en i ely ancho ed on e i iable c i e ia: unc ion g aph and lin-
ea ex ension, phase–densi y iden i y and NPE ini e-o de e o discipline, Legend e–B egman
and KKT op imiza ion s uc u e, he eby educing he one-dimensionali y o “na a i e ime”
o he esul o s uc u al choice + me ic eadou .
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