Unified Delay Theory of Entanglement--Consciousness--Time: Fourfold Bridging of Spectral--Scattering--Information--Discounting and Cross-Modal Verifiable Scales

Unied Delay Theo y o En anglemen ConsciousnessTime:
Fou old B idging o
Spec alSca e ingIn o ma ionDiscoun ing and C oss-Modal
Ve iable Scales
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
We p opose a unied delay heo y spanning om quan um sca e ing o conscious ime pe -
cep ion o social delay discoun ing. Fi s , we es ablish a scale iden i y based on spec al shi 
phaseg oup delay, p o iding egula iza ion ia Kon se ichVishik (KV) and ela i e de e mi-
nan s o coun able channels and inni e-dimensional cases, wi h explici almos e e ywhe e
die en iabili y and de-singula iza ion schemes in neighbo hoods o h esholds and embedded
eigens a es. Second, we cha ac e ize he con ac ion o local dis inguishabili y a e ia mono-
onici y o quan um Fishe in o ma ion, gi ing an ope a ional deni ion o subjec i e du a ion
and in oducing Pe z eco e y as he necessa y and sucien condi ion o equali y. Thi d, we
p o ide unied exp essions o eec i e ho izon wid h ia exponen ial, hype bolic, and quasi-
hype bolic discoun ing wi h mono onici y c i e ia, es ablishing discoun pa ame e mappings
h ough he  u u e-sel /o he -o e lap ac o . Fou h, we in oduce a la en coupling s eng h
pa ame e o c oss-align mic owa e sca e ing g oup delays, beha io al h esholds, and discoun
cu es ac oss laye s, p oposing join expe imen s and e o budge s. These h ee domains close
in enginee ing as a c oss-scale e iable amewo k o coupling enhancemen dwell inc ease
ho izon ex ension. Appendices p o ide de ailed p oo s o h esholdpole egula iza ion, QFI
equali y condi ions, discoun gene aliza ion, and comple e e o budge s.
Keywo ds
: Wigne Smi h g oup delay; Bi manK en spec al shi ; KV/ ela i e de e mi-
nan ; quan um Fishe in o ma ion; Pe z eco e y; subjec i e ime; delay discoun ing; hype bolic
and quasi-hype bolic; sel o he o e lap; mul i-laye s uc u al equa ions
1 In oduc ion & His o ical Con ex
The o ma ion o ime mani es s in h ee complemen a y languages: g oup delay and densi y o
s a es change in quan um sca e ing, dis inguishabili y a e and subjec i e du a ion dila ion in con-
scious ime pe cep ion, and discoun ing wi h eec i e ho izons in social decision-making. On he
sca e ing side, wo k by Wigne and Smi h es ablished he connec ion be ween phase de i a i es
and dwell imes, p ecisely dening g oup delay h ough he li e ime ma ix; he Bi manK en
o mula links he sca e ing de e minan o he spec al shi unc ion, o ming a closed loop o
phasespec al shi DOSg oup delay (wi h Ya ae 's amewo k as igo ous backg ound). In
sca e ing on ne wo ks and g aphs, F iedel summa ion can ail due o da k s a es, equi ing co -
ec ed coun ing o accessible channels.
On he consciousness side, quan um Fishe in o ma ion se es as he in insic me ic o pa-
ame e es ima ion, sa is ying mono onici y unde CPTP maps; i s equali y and equi alence wi h
1
 eco e abili y/sucien s a is ics a e sys ema ically cha ac e ized by Pe z heo y and subsequen
wo k. Human ime pe cep ion exhibi s e e sible  as slow dila ion phenomena unde modula ion
by emo ional and ewa d pa hways, wi h neu al mechanism e idence om classical e iews and
mul iple neu al su eys.
On he social side, empi ical ac s o delay discoun ing a e widely  ed as hype bolic o quasi-
hype bolic o ms (
β

δ
model), signican ly co ela ed wi h  u u e-sel con inui y/o he o e lap.
This pape unies he h ee domains unde he common scale o couplingdwellho izon, p o iding
c oss-modal join es s and e o closu e.
2 Model & Assump ions
Va iables and measu es
: Fix uni s
ℏ= 1
, uni o mly measu e in equency a iable
ω
; all de i a-
i es, densi y o s a es, and spec al shi a e aken wi h espec o
ω
.
Sca e ing-side assump ion (H
sca
)
: Sel -adjoin ope a o pai
(H, H0)
sa ises
H−H0∈
S1
o ela i e ace class; wa e ope a o s exis and a e comple e; ene gy-laye sca e ing ma ix
S(ω)
is die en iable excep on ze o-measu e se s o h esholdspolesembedded eigens a es. In
he inni e-dimensional case, dene
de KV S(ω)
and phase
Φ(ω) = a g de KV S(ω)
ia KV/ ela i e
de e minan s; he Koplienko case co esponds o second-o de spec al shi unde Hilbe Schmid
pe u ba ions. Th esholdpole de-singula iza ion is pe o med ia Jos unc ions and esol en
expansions, explici ly speci ying almos e e ywhe e die en iabili y and p incipal alue in eg al
in e p e a ion.
Ne wo k and channels (H
ne
)
: Channels decompose in o accessible subspace
Hacc
and da k
s a e subspace
Hda k
. All coun ing laws a e s a ed on
Hacc
, wi h local- o m co ec ions o DOS
when necessa y.
Consciousness side (H
cog
)
: Global e olu ion
ρAB(θ) = e−iθH ρABeiθH
, measu able only on
A
,
local channel
Λ = T B
. Quan um Fishe in o ma ion
FA
Q(θ)
is dened by mono one me ics (Pe z
class), sa is ying da a p ocessing inequali y and necessa y-sucien equali y condi ions.
Social side (H
soc
)
: Discoun weigh s adop unied weigh unc ion
V( )
, including exponen ial
V( ) = γ
, hype bolic
V( ) = (1 + k )−α
, and quasi-hype bolic
V(0) = 1, V ( ≥1) = βδ
. Dene
eec i e ho izon wid h
T∗=P ≥0w
(
w
no malized weigh ). Le  u u e-sel /o he o e lap index
be
C∈[0,1]
, mapping o model pa ame e s (e.g.,
γ, k, α, β, δ
) mono onically.
3 Main Resul s (Theo ems and Alignmen s)
Boxed econcilia ion (unied ac o h oughou )
: Fo any uni a y
S(ω)
and i s phase
Φ(ω) =
a g de S(ω)
, we ha e
Q(ω) = ∂ωΦ(ω)
, and
φ(ω) = 1
2Φ(ω)
. Thus he unied scale is
1
2π Q(ω) =
φ′(ω)
π=ρ el(ω)
(whe e
ρ el =−ξ′
). This equali y holds in inni e dimensions and unde ela i e
de e minan s, in e p e ed ia almos e e ywhe e de i a i es and de-singula iza ion egula iza ion.
Theo em 1
(Theo em 1: Applicabili y Domain and Regula iza ion o Spec al Shi PhaseG oup
Delay)
.
Unde (H
sca
), excep o ze o-measu e se s o h esholds/ esonances/embedded eigens a es,
o almos all
ω
we ha e
φ′(ω)
π=ρ el(ω) = 1
2π Q(ω)
. I
H−H0∈S2
a he han
S1
, eplace
wi h Koplienko spec al shi and second-o de de e minan o ob ain he co esponding second-o de
e sion. Th eshold neighbo hood co ec ions and da k s a e co ec ions a e in Appendix A.
Theo em 2
(Theo em 2: QFI Mono onici y and Equali y Condi ion o Local Time Scale)
.
Le
ρAB(θ)
and local channel
Λ = T B
. Fo any
FQ
co esponding o a Pe z mono one me ic, we ha e
FA
Q(θ)≤FAB
Q(θ)
. Equali y holds i and only i
Λ
is sucien o he s a e amily, i.e., he e exis s a
2
Pe z eco e y
Rσ
such ha
Rσ◦Λ[ρAB(θ)] = ρAB(θ)
(also holds o equi alen cha ac e iza ions such
as Rényi classes). This condi ion is he necessa y and sucien c i e ion o local non-dec ease.
P oposi ion 3
(P oposi ion 3: Ope a ionaliza ion o Subjec i e Du a ion and En anglemen Mono-
onici y)
.
Dene subjec i e du a ion
subj(τ) = Rτ
0(FA
Q( ))−1/2d
. I coupling/en anglemen en-
hancemen leads o
∂EFA
Q( )≤0
almos e e ywhe e, hen
∂E subj(τ)≥0
. Beha io al agen s a e
gi en by he quan um C amé Rao lowe bound
∆ min ≥[mFA
Q]−1/2
, so
(FA
Q)−1/2
can be es ima ed
om psychophysical h esholds.
Theo em 4
(Theo em 3: Dwell Law and A ea o Single Poles and Few Channels)
.
The B ei 
Wigne app oxima ion gi es
τg(ω) = Γ[(ω−ω0)2+ Γ2]−1
, wi h in eg al a ea
RRτg(ω)dω =π
; i
eedback educes eec i e decay
Γ = Γ(g)
mono onically dec easing wi h coupling, hen
τg(ω0) =
1/Γ(g)
inc eases mono onically. In mul i-channel cases, a ea is edis ibu ed by coupling weigh s.
Theo em 5
(Theo em 4: Ho izon Mono onici y o Exponen ial Discoun ing)
.
Le
V( ) = γ
and
T= (1 −γ)−1
. I
γ= Γ(C)
is s ic ly inc easing, hen
dT/dC = Γ′(C)/(1 −Γ(C))2>0
. This
mono onici y is consis en wi h e idence ha u u e-sel con inui y/o he o e lap enhances discoun
ac o s.
Theo em 6
(Theo em 5
′
: Eec i e Wid h Mono onici y o Hype bolic Discoun ing)
.
Le
V( ) =
(1 + k )−α
, ake
w =V( )/Ps≥0V(s)
, dene
T∗=P ≥0w
. Then
∂kT∗<0
and
∂αT∗>0
,
consis en wi h he exponen ial model in he
k→0
limi (co esponding o
γ→1−
). P oo in
Appendix D. Empi ically, hype bolic  s ou pe o m pu e exponen ial.
Theo em 7
(Theo em 5
′′
: Eec i e Wid h o Quasi-Hype bolic
β

δ
)
.
Le
V(0) = 1, V ( ≥1) =
βδ
. A e no maliza ion,
T∗= 1 + βδ/(1 −δ)
, so
∂βT∗>0
and
∂δT∗>0
. Me a-analyses show
β

δ
is he mains eam model o cha ac e izing p esen bias.
Theo em 8
(Theo em 6: C oss-Modal Iden iable Mapping ia La en Coupling S eng h)
.
In o-
duce la en a iable
κ
o uni o mly cha ac e ize coupling s eng h, assuming exis ence o mono one
die en iable mappings
Γphys :κ7→ Γ(g)
,
Γcog :κ7→ FA
Q
,
Γsoc :κ7→ (γ;k, α;β, δ)
. I join
expe imen s synch onously collec
τg(ω0),∆ min, γ
sa is ying a con ound- ee sepa able s uc u al
equa ion model, he co-di ec ionali y (sign consis ency) o he h ee mono onic ela ionships can
be es ed and he la en scale o
κ
es ima ed. P oo in Appendix E (iden ica ion condi ions and
es ima ion s a egy).
4 P oo s
P oo o Theo em 1 (key poin s)
: F om he Bi manK en o mula
de S(ω) = exp{−2πi ξ(ω)}
,
we ge
Φ′(ω) = −2πξ′(ω)
; and
Q=−iS†∂ωS
gi es
Q=−i ∂ωlog de S=∂ωΦ
. Thus
(2π)−1 Q=
ρ el =−ξ′=φ′/π
(
φ=1
2Φ
). Fo KV/ ela i e de e minan cases, eplace a he deni ion o
de
wi h KV ace and
ζ
- egula iza ion; h eshold and embedded eigens a es a e de-singula ized ia
Jos unc ions and esol en expansions, ensu ing almos e e ywhe e alidi y.
P oo o Theo em 2 (key poin s)
: Pe z-class mono one me ics sa is y da a p ocessing
inequali y o any CPTP map; le ing
Λ = T B
gi es
FA
Q≤FAB
Q
. Equali y is necessa y and
sucien o eco e abili y: he e exis s a Pe z eco e y
Rσ
such ha
Rσ◦Λ[ρAB(θ)] = ρAB(θ)
;
his condi ion can also be s a ed in Rényi amilies and
α

z
gene aliza ions.
P oo o P oposi ion 3 (key poin s)
: By he quan um C amé Rao lowe bound
∆ min ≥
[mFA
Q]−1/2
, we know
(FA
Q)−1/2
is he uni h eshold scale hickness. I
∂EFA
Q≤0
, hen
∂E subj(τ) =
Rτ
0
1
2(FA
Q)−3/2(−∂EFA
Q)d ≥0
.
3
P oo o Theo em 3 (key poin s)
: In B ei Wigne o m,
τg
is Cauchy densi y wi h a ea
π
. I
Γ′(g)<0
, hen
∂gτg(ω0) = −Γ′/Γ2>0
. Mul i-channel decomposi ion by coupling, a ea
dis ibu ion gi en by pa ial wid hs.
P oo s o Theo ems 45
′
5
′′
(key poin s)
: Exponen ial case yields mono onici y by di ec
die en ia ion. Hype bolic case uses in eg al ial
P ≥0(1+k )−α≈R∞
0(1+k )−αd =k−1(α−1)−1
(
α > 1
); a e no maliza ion,
T∗
is mono one wi h
k↓
and
α↑
. Quasi-hype bolic yields closed o m
by di ec summa ion wi h mono onici y. Model  supe io i y in ele an e iews and me a-analyses.
P oo o Theo em 6 (key poin s)
: Assume obse a ion iple
(τg(ω0),∆ min, γ)
is gen-
e a ed by h ee mono one mappings o la en a iable
κ
wi h addi i e noise, s uc u al equa ions
Yj=hj(κ) + εj
. I
hj
a e mono one and noise independen , use ank co ela ion consis ency and
mul i-le el SEM o es ima e co-di ec ionali y o
sign(∂hj/∂κ)
; c oss-modal co-di ec ionali y is he
e iable c i e ion o unied coupling hypo hesis. Iden ica ion de ails in Appendix E.
5 Model Applica ions
Physical sidechannel/ne wo k dwell measu emen
: On a wo-po ec o ne wo k pla o m,
measu e
S(ω)
and use obus unw apping and Cauchy smoo hing die ence o compu e
φ′(ω)
and
Q(ω)
, e i ying
(2π)−1 Q=φ′/π
and consis ency wi h DOS coun ing; in eedback ca i ies,
une
g
o  mono one law o
Γ(g)
, epo ing a ea conse a ion. Fo non-minimum phase loops, use
Bode/K ame sK onig ela ions o phasemagni ude consis ency checks.
Consciousness sideslow hick subjec i e du a ion
: Time ep oduc ion and minimum
die ence h esholds in pa allel, es ima ing
∆ min
and subjec i e a ings; in high-connec ion con-
ex s, expec
∆ min ↑
,
subj ↑
, aligned wi h in e nal clockdopamine modula ion e idence.
Social sidediscoun ingho izon
: Use adap i e i a ion o ob ain indi idual
γ
o
(k, α)
/(
β, δ
),
synch onously collec IOS and u u e-sel con inui y, e i ying mono one mappings o Theo ems 4
5
′
5
′′
.
6 Enginee ing P oposals
P1 | Canonical measu emen o mic owa e ne wo k g oup delay
: Phase unw apping h esh-
old se ing, equency g id
∆ω
, equi alen noise bandwid h and po misma ch e o budge ing; use
h ee-poin / e-poin die ence and spline de i a i e c oss- alida ion; o non-minimum phase, co -
ec pa asi ic phase ia Bode gainphase ela ions and Hilbe ans o m.
P2 | Subjec i e du a ionQFI p oxy dual- ask pa adigm
: Oddball e en induc ion and
neu al sequence con ol in pa allel, collec
∆ min
, pupil/skin conduc ance/HRV o mul i-modal
usion o isola e a ousal con ounds; map
∆ min
o
Fbeh
Q∝∆ −2
min
ia CRB.
P3 | Hie a chical Bayesian  o discoun cu es
: Simul aneously  exponen ial/hype bolic/quasi-
hype bolic and compa e wi h WAIC/AIC/BIC; collec IOS and u u e-sel con inui y o media ion
analysis; p osocial/ us manipula ions unde e hical compliance as ex e nal alida ion.
7 Discussion (Risks, Bounda ies, Pas Wo k)
Applicabili y domain
: The iden i y holds wi hin
S1
o ela i e ace class; Koplienko case gi es
second-o de e sion; h esholdsembedded eigens a es equi e Jos / h eshold expansion ea men ;
da k s a es in g aphs and eedback ne wo ks need explici exclusion o local DOS co ec ion.
P o able es able bounda ies
: On he consciousness side, subjec i e du a ion as beha -
io al p oxy o
F−1/2
Q
elies on nea -sa u a ion o CRB; expe imen al es s o nea -sa u a ion
4
and bias co ec ion. Social-side model he e ogenei y con olled ia hie a chical Bayes and model
compa ison.
Rela ionship wi h exis ing wo k
: This amewo k ancho s in spec alsca e ing scales,
aligning quan ica ion o conscious and social ime on he common geome y o dwelldis inguishabili y
ho izon; i does no claim me aphysical iden i y o  ime equals en anglemen , bu p o ides c oss-
domain ope a ional equi alence and join c i e ia. See e e ences o classical e iews and mode n
ad ances.
8 Conclusion
On he igo ous ounda ion o spec alsca e ing egula iza ion and QFI mono onici y, we p o-
ide unied scales and mono one laws o subjec i e du a ion and social ho izons, es ablishing he
e iable iad o coupling enhancemen dwell inc easeho izon ex ension. The common la en
a iable scale ac oss h ee domains b ings mic owa e sca e ing, beha io al h esholds, and discoun
cu es in o he same s a is icalcausal s uc u e, cons i u ing an enginee ing pa h o c oss-scale
ime heo y.
Acknowledgemen s, Code A ailabili y
Thanks o publicly a ailable li e a u e on spec al shi g oup delay, quan um Fishe in o ma ion,
and delay discoun ing. Sc ip s o
S
-pa ame e  ing, phase unw apping, CRB es ima ion, and
hie a chical Bayesian discoun  ing can be di ec ly implemen ed om appendix pseudocode.
Re e ences
Wigne (1955); Smi h (1960) g oup delay and li e ime ma ix; Ya ae (1992/2010) sca e ing heo y
monog aph; Pushni ski (2010) Bi manK en; Texie (2001/2003) F iedel on g aphs; Kon se ich
Vishik (1994) KV de e minan ; Gesz esyPushni skiSimon (2007) Koplienko; Pe z (1996/1988)
mono one me ics and eco e y; Eagleman (2008) ime pe cep ion e iew; E sne -He sheld (2009/2011)
u u e-sel con inui y; Mazu (1987) hype bolic discoun ing; Laibson (1997)
β

δ
.
A Rigo ous Iden i y in Inni e Dimensions and a Th esholds
Poles
A.1 KV/ ela i e de e minan and almos e e ywhe e de i a i e
: Le
de KV S(ω) =
exp{TRKV log S(ω)}
, phase
Φ = a g de KV S
. In
S1
case, Bi manK en gi es
Φ′=−2πξ′
; in
S2
case, adop Koplienko spec al shi o dene second-o de equali y.
Φ′
exis s excep on se s o
h esholdspolesembedded eigens a es.
A.2 Jos / h eshold expansion
: Th eshold neighbo hoods adop Jos unc ions and esol en
expansions, cla i ying p incipal alue in e p e a ion and addi ional e ms o
ξ′
; embedded eigen-
s a es leak ia Fe mi golden ule in o egula pa e ns, see JensenKa o and subsequen h eshold
expansion wo k.
A.3 Sca e ing on g aphs and da k s a e co ec ion
: Fo local s a es in
Hda k
, F iedel
coun ing needs o sub ac con ibu ions om inaccessible s a es; local DOS and injec ion/emission
a es gi e local e sion.
5

B QFI Mono onici y, Equali y Condi ion, and Subjec i e Du a ion
B.1 Da a p ocessing and equali y
: Pe z esul s show ha o mono one me ic
FQ
and channel
Λ
,
FQ(Λ[ρθ]) ≤FQ(ρθ)
; equali y i and only i
Λ
is sucien o he s a e amily, eco e y
Rσ
exis s.
Equali y c i e ia o Rényi and
α

z
ex ensions a e equi alen o Pe z eco e y.
B.2 Subjec i e du a ion
: Take
subj(τ) = Rτ
0(FA
Q( ))−1/2d
. CRB gi es
∆ min ≥[mFA
Q]−1/2
;
subs i u ing beha io al measu e o
∆ min
yields empi ical es ima ion o mula; es ima e is unbiased
when nea -sa u a ion op imal measu emen exis s.
C A ea Conse a ion o Single Poles and Few Channels
B ei Wigne
τg
is s anda d Cauchy o m, a ea
π
independen o
Γ
; in mul i-channel coupling, a ea
is dis ibu ed by pa ial wid hs; in ca i y eedback ne wo ks, es coupling enhancemen dwell
inc ease by  ing mono one ela ion o
Γ(g)
and
τg(ω0)
.
D Discoun Gene aliza ion and Eec i e Wid h
D.1 Hype bolic amily
:
V( ) = (1 + k )−α
, no maliza ion cons an
Z=P ≥0(1 + k )−α≈
k−1(α−1)−1
; eec i e wid h
T∗=Pw
mono one wi h
k↓
and
α↑
.
D.2 Quasi-hype bolic
β

δ
:
T∗= 1 + βδ/(1 −δ)
; when
β→1
,
δ→γ
e u ns o exponen ial.
Li e a u e compa ison shows hype bolic and quasi-hype bolic ou pe o m pu e exponen ial ac oss
mul iple ca ego ies.
E Join Iden ica ion and S a is ical Powe
E.1 S uc u al equa ions and iden ica ion
: Le
τg(ω0) = h1(κ)+ε1,∆ min =h2(κ)+ε2, γ =
h3(κ) + ε3
,
hj
mono one,
εj
independen . Use mul i-le el SEM and ank co ela ion o es co-
di ec ionali y o
sign(h′
1),sign(h′
2),sign(h′
3)
.
E.2 S a is ical powe and sample size
: Ta ge eec size
d∈[0.3,0.5]
de ec ion equi es
ens o hund eds o samples; epo mul iple compa ison co ec ion and pos -hoc manipula ion es s.
Powe and eec size epo ing ollows s anda d guidelines.
F Implemen a ion De ails and E o Budge
F.1 Mic owa e ne wo k (P1) e o closu e
: (i) Phase unw apping: se allowable jump h esh-
old and esidual de ec ion; (ii) die ence and spline de i a i e c oss- alida ion, Cauchy smoo hing
die ence supp esses high- equency noise; (iii) po misma ch and ENBW co ec ion; (i ) non-
minimum phase de ec ion and Bode/Hilbe co ec ion.
F.2 Time pe cep ion (P2) and CRB mapping
: Pa allel collec ion o
∆ min
, subjec i e
a ings, and physiological indica o s, isola ing a ousal/a en ion con ounds; map
∆ min
o
Fbeh
Q
ia
CRB, epo ing condence in e als.
F.3 Discoun ing (P3) model compa ison
: Exponen ial/hype bolic/quasi-hype bolic simul-
aneously  ed wi h WAIC/AIC/BIC; hie a chical Bayes mi iga es indi idual he e ogenei y; collec
IOS and u u e-sel con inui y as explana o y a iables o media ion eg ession; e hical and blinding
con ols o po en ial p osocial/ us manipula ions.
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