Time Equivalence Class and Generalized Entropy Optimization:\\ Unified Rates, Rigorous Axioms, and Observation-Oriented Closed Framework

Time Equi alence Class and Gene alized En opy
Op imiza ion:
Unied Ra es, Rigo ous Axioms, and
Obse a ion-O ien ed Closed F amewo k
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
We p opose and igo ize a unied amewo k cen e ed on he ime equi alence
class
[T]
, placing he a ow o ime, black hole in o ma ion, cosmological ed-
shi /cons an , and measu able delayphasespec al shi  on a common compu-
a ional/obse a ional pla o m. Fi s , we p o ide a p ecise deni ion o
[T]
, p o e
eexi i y, symme y, and ansi i i y o he equi alence ela ion, and cla i y co-
a iance c i e ia wi h espec o cu amilies and p ese a ion unde s a e/ egion
changes. Second, we es ablish he
unied a e iden i y
ρ el(ω) = 1
2π∂ωϕ(ω) = 1
2πT Q(ω),
whe e
Q(ω) = −i S†(ω)∂ωS(ω)
is he Wigne Smi h ime delay ope a o ,
ϕ(ω) =
a g de S(ω)
is he o al sca e ing phase,
ρ el(ω)
is he spec al densi y dened by
spec al shi / ela i e en opy; his iden i y a ises om Bi manK en ela ions and
obse able phasedelay measu emen s, wi h consis en dimensions and in e ibili y.
Thi d, in he semiclassical egime wi h Hadama d s a es and weak cu a u e, ia he
chain  ela i e en opy mono onici y
⇒
QNEC
⇒
local GSL/QFC, we p o e: along
a null geodesic amily wi h ane pa ame e
λ
,
Sgen
is mono onic; epa ame iza-
ion by
[T]
gi es
Sgen
mono onici y wi h espec o ep esen a i e ime
, hus
he a ow o ime eme ges as an ou pu p ope y. Fou h, using algeb aic embed-
ding/en anglemen wedge language, we show xed-p ojec ion non-decodabili y
≡
ime-map singula i y a e en ho izon, and ealize analy ic con inua ion h ough
ex emal swi ching o he island o mula, he eby eco e ing he Page cu e. Fi h,
we iew
Λ
as a global calib a ion in eg a ion cons an o
[T]
(compa ible wi h ou -
o m mechanisms), emphasizing i s dis inc ion om he measu able eec o local
 acuum ene gy densi y. Finally, we p o ide h ee ope a ional e ica ion pa h-
ways: dispe siongeome y coupled ime delay o o de
ω−2
in cu ed space ime
plasma geome ical op ics (wi h null- es p o ocol), hie a chical Bayesian es o
 edshi decohe ence slope o FRBs, and ime-delay Bell wi ness and chi al spli -
ing in c yogenic mul i-mode ca i ies. Appendices include: comple e p oo s o h ee
equi alence p ope ies; ope a o spec al shi de i a ion o he unied a e iden i y;
1
p oo o main heo ems d i en by QNEC/GSL; cu ed space imeplasma eikonal ex-
pansion (showing condi ions o absence o
ω−1
p incipal e m); gauging, s abili y,
and low-ene gy cons ain s o ime-eld heo y.
Keywo ds:
Time Equi alence Class; Modula Time; Gene alized En opy; QNEC; QFC;
QES/Island Fo mula; Wigne Smi h Time Delay; Bi manK en Spec al Shi ; Plasma
Geome ical Op ics; Hie a chical Bayesian

1 In oduc ion and His o ical Con ex
The gene alized en opy
Sgen =A/4Gℏ+Sou
in semiclassical quan um g a i y connec s
geome y and in o ma ion. Non-pe u ba i e econs uc ion o he Page cu e elies on
quan um ex emal su aces and he island o mula; QNEC and (local) QFC ela e null-
di ec ional en opy de o ma ion o s ess-ene gy enso s and quan um ocusing, and can
be de i ed om ela i e en opy mono onici y. On he o he hand, Tomi aTakesaki
modula heo y iews he modula ow
σω
s
o a s a ealgeb a pai
(A, ω)
as in in-
sic ime, p o iding ma hema ical suppo o he pe spec i e ha  ime eme ges om
in o ma ioncausal s uc u e. On he sca e ing side, he Wigne Smi h delay ope a o
and o al sca e ing phase
ϕ
a e di ec ly measu able, and he K en spec al shi unc-
ion
ξ(ω)
couples wi h
de S(ω)
ia he Bi manK en o mula. This pape closes hese
suppo ing poin s in o a unied whole: go e ning  ime h ough he equi alence class
s uc u e o
[T]
, and b inging phasedelayspec al shi in o a common expe imen ally
accessible gauge h ough he
unied a e
.

2 Model and Assump ions
2.1 Time Equi alence Class and Co a iance
Objec s
: Globally hype bolic
(M, g)
; egion amily
R
; s a ealgeb a pai s
{(A(R), ωR)}R∈R
;
cu clus e o null geodesic amilies
C
.
Deni ion 2.1
(Equi alence (Deni ion 2.1))
.
Fo xed
(A(R), ωR,C)
, we say
T1∼T2
i he e exis an ou e au omo phism
Φ∈Ou (A(R))
and a s ic ly mono onic
:R→R
such ha
Φ◦σωR
s◦Φ−1=σωR
(s)
and
E(C;T1) = E(C;T2),
whe e
E
ep esen s he gene alized en opy mono onici y/ex emal s uc u e (quan um
expansion signa u e and null se ) along null gene a o s o
C
.
P oposi ion 2.2
(Th ee P ope ies (P oposi ion 2.2))
.
∼
is an equi alence ela ion; o
comple ely posi i e ace-p ese ing maps induced by s a e/ egion homo opy de o ma ions,
i he ou e conjugacy class o ela i e modula ow is in a ian , hen
[T]
is p ese ed.
Unde cu amily enemen , i
Θq≥0
, hen
E
is p ese ed, and
[T]
is said o be co a ian
unde ha null geodesic amily. P oo in Appendix A.1A.2.
2
2.2 Unied Ra e and Dimensional Consis ency
Wigne Smi h and Phase
:
Q(ω) = −i S†∂ωS
,
T Q=∂ωa g de S=∂ωϕ
.
Spec al Shi and Phase
: Bi manK en:
de S(ω) = e−2πi ξ(ω)⇒∂ωϕ=−2π ξ′(ω)
.
Unied Ra e
:
ρ el(ω) = −ξ′(ω) = 1
2π∂ωϕ(ω) = 1
2πT Q(ω).
The deni ion o
ρ el
adop s local window a ia ion, so ha
S(ρ|σ) = Rρ el(ω) dω
;
dimensions a e consis en wi h
∂ωϕ
,
T Q
. Nume ical in e sion is egula ized using
K ame sK onig and phase unw apping. De i a ion and in e sion de ails in Appendix
A.3.
2.3 Domain o Assump ions and Failu e Condi ions
Hadama d s a es, weak cu a u e, local Rindle app oxima ion and con olled de o ma-
ions; geome ic congu a ions es ic ed o quan um ligh shee s/e en ho izons and
o he GSL-applicable si ua ions. Failu e domains: s ong cu a u e neighbo hoods, non-
Hadama d s a es, UV-domina ed de ia ions, e c.

3 Main Resul s (Theo ems and Alignmen s)
3.1 Theo em 3.1 (A ow o Time = Ou pu o Gene alized En-
opy Mono onici y)
Wi hin he domain o Sec ion 2.3, along a specied quan um ligh shee wi h ane
pa ame e
λ
:
dSgen/dλ≥0
. Fo any
∈[T]
, i
= (λ)
wi h
′>0
, hen
dSgen/d ≥0
.
P oo chain and ailu e condi ions de ailed in Appendix B.
3.2 Theo em 3.2 (Black Hole In o ma ion: Fixed-P ojec ion Non-
Decodabili y and Island Fo mula Analy ic Con inua ion)
Using algeb aic embedding
A(I+)⊂ A(D)
o exp ess ex e nal obse a ions; xed p ojec-
ion co esponds o i e e sible CPTP dimensionali y educ ion maps, in o ma ion loss
is me ely non-decodabili y unde ha p ojec ion. Island o mula h ough saddle-poin
swi ching (QES) is equi alen o analy ic con inua ion wi hin
[T]
, expanding he econ-
s uc able domain
⇒
Page cu e eco e y. Appendix C p o ides explici isomo phism in
JT scena ios.
3.3 P oposi ion 3.3 (Redshi = Time Uni Rescaling)
(1 + z)=(k·u)e/(k·u)o
is a co a ian exp ession o  hy hm a io, equi alen o FRW
a0/ae
. The inc emen is mani es ed in he unied a e di ec ly ela ing his a io o
obse a ional in e sion o
ϕ′(ω)
and
ρ el
.
3
3.4 Theo em 3.4 (Time Holog aphy and Time Quan um E o
Co ec ion)
JLMS and en anglemen wedge nes ing imply:
[T(p)] = Π(s, γp)
; he p ojec ion amily o
[T]
sa ises KnillLaamme condi ions on code subspace
C
, hus  ime-selec ion e o s
can be co ec ed by equi alence class edundancy. Appendix D p o ides modula Be y
cu a u e and measu able phases o pa h dependence.
3.5 Theo em 3.5 (
Λ
as GaugeIn eg a ion Cons an and Fou -
Fo m Disc e iza ion)
T ace- ee/Unimodula and ou - o m mechanisms make
Λ
an in eg a ion cons an ; in
he he mal ime gauge i s seman ics is a global calib a ion o
[T]
, compa ible wi h nea -
disc e e spec um induced by axion ou - o m quan iza ion. Appendix E p o ides ac ion
and a ia ion.

4 P oo s
4.1 A ow o Time (Theo em 3.1)
Rela i e en opy mono onici y
⇒
ANEC/QNEC; combined wi h quan um Raychaudhu i
gi es quan um expansion non-inc ease in null di ec ion,
Sgen
mono onici y. S ic mono-
onici y o
= (λ)
gi es mono onici y o any ep esen a i e ime. Failu e domains
include s ong cu a u e and non-Hadama d s a es. De ails in Appendix B.
4.2 Coo dina e-Independen Fo mula ion o Black Hole In o ma-
ion (Theo em 3.2)
Cha ac e ize non-decodabili y using ela i e en opy and Pe z eco e y; ex emal swi ch-
ing o island o mula is equi alen o changing ep esen a i e o
[T]
and expanding en-
anglemen wedge. In JT model, demons a e his equi alence using eplica geome y.
4.3 Unied Ra e (Equa ion 2.2)
Bi manK en:
de S=e−2πiξ ⇒∂ωϕ=−2πξ′
; mul i-channel
T Q=∂ωϕ
. Dene
ρ el =−ξ′
o ob ain he s a ed iden i y. Nume ical in e sion con ols noise amplica ion
using phase unw apping and K ame sK onig egula iza ion.

5 Model Applica ion: Two Minimal Models
5.1 1+1 Dimensional CFTRindle
Single-channel sca e ing
S=eiϕ ⇒Q=∂ωϕ
; decompose ela i e en opy in o spec al
window in eg als o e i y
ρ el = (2π)−1∂ωϕ
. KMS scale consis en wi h modula ow
escaling.
4
5.2 JT G a i y + F ee Field
Page ansi ion co esponds o QES saddle-poin swi ching; analy ic con inua ion in
[T]
con e s ex e nal p ojec ion non-decodabili y o expanded econs uc ion domain de-
codabili y. Appendix C p o ides equa ion amily and schema ic cu es.

6 Enginee ing P oposals (Magni udes, Sys ema ics, and
Null-Tes s)
6.1 Deep Space Mul i-F equency Links: Time Delay in Cu ed
Space imePlasma Eikonal
Geome ical Op ics
: S a ic weak eld
Φ/c2≪1
, iso opic plasma:
n(ω, x) = q1−ω2
p(x)/ω2,d
dℓ≃1
c1 + 2Φ
c21 + ω2
p
2ω2.
Pa h a ia ion gi es
∆ (ω)=∆ Shapi o +Zω2
p
2ω2
dℓ
c+ZΦ
c2
ω2
p
ω2K(x)dℓ
c+O(ω−4).
Conclusion
: Dominan coupling e m is
ω−2
a he han
ω−1
;
ω−1
only possible in
aniso opic media o s ong non-adiaba ic uids.
Magni udes and Sys ema ics
: Ka/X (832 GHz) a impac angles
5◦−15◦
can
achie e picosecond-le el die ences; main sys ema ic e o s a e ionosphe ic/heliosphe ic
modeling and ha dwa e nonlinea i y.
Null-Tes s
: Geome ic commu a ion (impac angle ip) should p ese e
ω−2
scaling
while changing geome ic weigh sign; day-nigh die ence o same geome y cancels
ionosphe ic p incipal e m.
Facili ies
: DSN X/Ka and DSAC s abili y links.
6.2 FRB Redshi Decohe ence Slope Hie a chical Bayesian
Model
:
W∼W0(1 + z)α(ν/ν0)−β
, hos /IGM/ins umen componen s in hie a chical
p io s; selec ion unc ion based on channeliza ion h eshold and DM
z
in e sion unce -
ain y.
Powe
: Ca alog-1 and subsequen localized samples a
N∼102−103
can dis inguish
he null hypo hesis
α= 0
a
5σ
; null- es wi h andomized
z
should wash ou slope.
6.3 C yogenic Mul i-Mode Ca i y: Time Delay Bell Wi ness and
Chi al Spli ing
BellTime Delay Wi ness
:
W=|T QA⊗T QB−T QAB|
, classical
≤0
, quan um
coupling
W>0
.
Chi al Spli ing
: Con olled mic o-dis o ion and g a i yelec omagne ic weak
coupling cause
∆ chi al ∝ω
linea spli ing.
5

Noise Budge
: Con ibu ions om phase whi e noise, TLS
1/
, he mally induced
equency shi s, mechanical mic ophonics, coun ing dead ime, e c., and a ge sub-
picosecond sensi i i y a e p o ided in Appendix G wi h o mulas/ ables.

7 Time-Field Theo y: Gauging, S abili y, and Low-
Ene gy Cons ain s
In oduce clock 1- o m
uµ=∂µT
√−∂αT ∂αT,
cons uc minimal ac ion
LT=M2
T
2hc1∇µuν∇µuν+c2(∇µuµ)2+c3aµaµi+V(T) + Lma e (ψ;uµ),
using S ückelbe g o handle epa ame iza ion edundancy
T7→ (T)
. Ghos - ee and
causally s able domain, PPN and g a i a ional Che enko cons ain s gi e easible sub-
space o
(ci)
; key dis inc ion om æ he /kh onon is: he p e e ed di ec ion he e is
me ely a gauge ep esen a i e o equi alence class, obse abili y concen a ed on a e
in a ian s
∂ωϕ
,
T Q
a he han aniso opy.

8 Discussion (Consis ency, Bounda ies, and Connec-
ions)

Consis ency
: Main heo ems a e s ic ly es ic ed o Hadama d/weak cu a u e
and quan um ligh shee /e en ho izon geome y; ou side he domain, only conjec-
u e s eng h is main ained.

Black Hole In o ma ion
: Algeb aicisland o mula isomo phism a oids coo di-
na e dependence; JT scena ios p o ide checkable ins ances.

Λ
: As a gauge in eg a ion cons an , no equal o  acuum ene gy densi y; compa -
ible wi h ou - o m disc e iza ion/axion mechanisms.

Ve iabili y
: Deep space links and FRB pipelines p o ide ep oducible imple-
men a ion elemen s and null- es s; c yogenic ca i y expe imen s p o ide indoo
epea able e ica ion pla o ms.

9 Conclusion
We es ablish he igo ous equi alence s uc u e o
[T]
and he unied a e, p o ide com-
mon seman ics o a ow o ime, black hole in o ma ion, edshi , and
Λ
, and g ound
he unica ion o  imecausali yin o ma ion a he da a le el h ough implemen able
6
obse a ional/expe imen al pa hways. This amewo k closes unde he iple c i e ia o
p o ablequan iable e iable.

Acknowledgemen s, Code A ailabili y
Thanks o public li e a u e and ma e ials on QNEC/QFC, QES/island o mula, modula
heo y, spec al shi sca e ing heo y, and cu ed space ime plasma geome ical op ics.
Appendices p o ide in e sion om phase da a o
ρ el
and minimalis implemen a ion
p o ocol o FRB hie a chical Bayesian.

Re e ences
[1] Bousso, Fishe , Leichenaue , Wall. Quan um ocussing and inequali ies,
Phys.
Re . D
93 (2016).
[2] Faulkne e al. Modula Hamil onians o de o med hal -spaces and he ANEC,
JHEP
(2016).
[3] Engelha d & Wall. Quan um ex emal su aces,
JHEP
01 (2015) 073.
[4] Almhei i e al. Replica wo mholes and he en opy o Hawking adia ion,
JHEP
05 (2020) 013.
[5] Connes & Ro elli. Von Neumann algeb a au omo phisms and he he mal ime
hypo hesis, (1994).
[6] Smi h. Li e ime ma ix in collision heo y,
Phys. Re .
118 (1960). (Wigne 
Smi h)
[7] Bi man & K en. On wa e and sca e ing ope a o s,
So . Ma h. Dokl.
(1962).
[8] Pe lick. Ray op ics in a plasma on a GR space ime, and ela ed wo ks.
[9] Bisno a yi-Kogan & Tsupko. G a i a ional lensing in plasma, e iews (2015
2022).
[10] Roge s. F equency-dependen lensing in plasma,
MNRAS
451 (2015).
[11] JLMS and successo s: bounda ybulk ela i e en opy equi alence.
[12] S anda d cosmology ex s o edshi p ojec ion iden i y.
[13] DSN/DSAC capabili y b ie s.
[14] CHIME/FRB Collabo a ion. The  s ca alog,
ApJS
257 (2021).

7
A P oo s o Equi alence Rela ion and Unied Ra e
A.1 Th ee P ope ies and Ou e Conjugacy (A.1)
G oup ac ion o
G= Ou (A)⋊Di +(R)
and o de p ese a ion gi e eexi i ysymme y
ansi i i y; p ese a ion unde s a e/ egion homo opy classes gua an eed by ou e con-
jugacy in a iance.
A.2 Cu Family Co a iance (A.2)
Wi hin QNEC sucien -necessa y domain, cu enemen co esponds o subalgeb a e-
s ic ion,
Sgen
o de p ope y p ese ed.
A.3 Bi manK en
⇒
Unied Ra e (A.3)
de S=e−2πiξ ⇒∂ωϕ=−2πξ′
; mul i-channel
T Q=∂ωϕ
, yielding
ρ el =−(ξ′) =
(2π)−1T Q
. Phase unw apping and K ame sK onig p o ide obus in e sion.
B Rela i e En opy
⇒
QNEC
⇒
Local GSL Fo m
B.1
Da a p ocessing inequali y and subalgeb a es ic ion;
B.2
Second-o de o mula o modula Hamil onian unde hal -space de o ma ion de-
i es ANEC/QNEC;
B.3
Quan um Raychaudhu i combined gi es
Θq≤0
, hence
dSgen/dλ≥0
.
C Algeb aic P oposi ion o Black Hole In o ma ion and
JT Example
C.1
Fixed p ojec ion co esponds o i e e sible CPTP, Pe z eco e y quan ies non-
decodabili y;
C.2
Island o mula saddle-poin swi ching equi alen o changing ep esen a i e in
[T]
and expanding en anglemen wedge;
C.3
JT scena io: eplica geome y saddle-poin swi ching example diag am o spec-
al window p ojec ion o
ϕ′(ω)
.
D Modula Be y and Time Quan um E o Co ec-
ion
D.1
[T(p)] = Π(s, γp) = P expRγpAmod
;
D.2
KnillLaamme condi ion sa is ac ion and h eshold unde  ime-selec ion e o 
noise model.
E
Λ
as GaugeIn eg a ion Cons an and Fou -Fo m
E.1
Ac ion and a ia ion;
8
E.2
Nea -disc e e spec um unde quan iza ion/axion coupling;
E.3
Seman ic co espondence wi h equi alence class calib a ion ze o poin .
F FRB Hie a chical Bayesian Pipeline (Rep oducible
Implemen a ion Skele on)
Minimalis ow o likelihood, p io s, selec ion unc ion, pos e io -p edic i e, and null-
es s.
G C yogenic Ca i y Expe imen Noise Budge
Con ibu ion o mulas, ep esen a i e pa ame e s, and congu a ion able o achie ing
sub-picosecond o phase whi e noise, TLS
1/
, he mally induced equency shi s, me-
chanical mic ophonics, eadou dead ime.

Symbol Table
[T]
;
σω
s
;
ϕ(ω)
;
Q(ω)
;
ρ el
;
Θq
;
λ
;
C
.
9