Framework of Limit Unification and No-Observer Ontologization\\ \large Unified Time Scale, Boundary Time Geometry, and Consistent Variational Principle

F amewo k o Limi Unica ion and No-Obse e
On ologiza ion
Unied Time Scale, Bounda y Time Geome y, and Consis en Va ia ional
P inciple
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 19, 2025
Abs ac
In p e ious wo ks, we ha e cha ac e ized he Uni e se as a maximal, consis-
en , and comple e on ological ma hema ical objec , in e nally con aining mul iple
laye s o componen s such as causal mani olds, unied ime scale, bounda y ime
geome y, bulk quan um eld heo y, sca e ing and spec al shi heo y, Tomi a
Takesaki modula s uc u e, and gene alized en opy. Unde his amewo k, an
in eg a ed desc ip ion o ime, causali y, en opy, and obse a ion can be achie ed,
bu physical laws hemsel es and physical de ails (gauge g oups, eld con en ,
mass spec um and couplings, uid and many-body eec i e equa ions, e c.) mos ly
appea as ex e nal addi ions.
This pape p oposes a amewo k o **limi unica ion comple ely independen
o any obse e on ological concep **: we ake unied ime scale and bounda y
ime geome y as he sole undamen al geome icspec al s uc u es, uni ying all
physical laws in o necessa y condi ions o a single consis en a ia ional p inciple.
Specically:
1. In oducing he scale iden i y in sca e ing heo y
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),
(1)
uni ying sca e ing phase de i a i e, ela i e densi y o s a es, and Wigne Smi h
g oup delay ace in o a unique ime scale mo he ule .
2. In oducing o al connec ion in bounda y ime geome y
Ω∂=ωLC ⊕AYM ⊕Γ es,
(2)
uni ying g a i y, in e nal gauge elds, and esolu ion/ eno maliza ion g oup ow
in o a single geome ic objec on he bounda y bundle.
3. In oducing gene alized en opy on small causal diamonds
Sgen(D) = A(∂D)
4Gℏ+Sbulk(D),
(3)
and cons uc ing a global consis ency unc ional
I[U] = Ig a +Igauge +IQFT +Ihyd o,
(4)
1
whe e each e m cons ains consis ency a geome icen opy, gauge opological,
quan umsca e ing, and coa se-g ained uid le els espec i ely.
The main esul o his pape is: unde na u al assump ions o causali y, uni a -
i y, and en opy s abili y, applying he unied consis ency p inciple
δI[U] = 0
(5)
o he Uni e se On ology Objec
U
, he necessa y condi ions a e espec i ely equi -
alen o:
1. Geome ic a ia ion in he limi o small causal diamonds yields Eins ein
equa ions
Gab + Λgab = 8πG⟨Tab⟩,
(6)
along wi h app op ia e quan um ene gy condi ions and ocusing condi ions;
2. Unde xed
K
- heo y class condi ions, a ia ion o bounda y channel bundle
and o al connec ion yields YangMills equa ions and gauge eld anomaly cancel-
la ion condi ions, he eby uni ying eld con en and gauge g oups as consis ency
equa ions o bounda y
K
class and sca e ing
K1
class;
3. Unde gi en geome ic and gauge backg ound, a ia ion o ela i e en opy
unc ional o bulk s a es and sca e ing da a yields Wigh man axioms, Eule 
Lag ange eld equa ions, and Wa d iden i ies o local quan um eld heo y, meaning
QFT is no longe an independen inpu bu an ine i able esul o he unied con-
sis ency p inciple a he quan umsca e ing le el;
4. In long-wa eleng h and low- esolu ion limi s, a ia ion o esolu ion connec-
ion and mac oscopic conse ed cu en s yields gene alized Na ie S okes ype uid
equa ions and diusion equa ions, uni ying mac oscopic i e e sible dynamics as
g adien ows o gene alized en opy on he unied ime scale.
The e o e, wi hou in oducing any obse e on ology concep , his pape achie es
limi unica ion in physics: Gene al Rela i i y, Gauge Field Theo y, Local Quan-
um Field Theo y, Fluid Dynamics, and Many-Body Eec i e Dynamics a e all
necessa y condi ions o he same cosmic consis ency a ia ional p inciple unde di -
e en esolu ions and bounda y condi ions, while all physical de ails a e uniedly
encoded in bounda y
K
- heo y classes and sca e ing analy ic in a ian s.
1 In oduc ion
1.1 Unica ion P oblem and Residual Deg ees o F eedom
Majo heo ies o mode n physicsGene al Rela i i y, Quan um Field Theo y, S a is ical
Physics and Fluid Dynamics, Condensed Ma e and Many-Body Sys emsha e been
ully e ied a hei espec i e applicable scales. Howe e , when a emp ing o p o ide
a unied heo y o he uni e se, a undamen al dicul y pe sis s:
1. We can uni y space ime and causali y a he geome ic le el (causal mani olds,
Lo en zian geome y);
2. We can uni y a ious in e ac ions a he quan um le el in o gauge eld heo y
sys ems (YangMills + Higgs + e mions);
3. We can link en opy, ene gy condi ions, and ime a ows a he in o ma ion le el
(gene alized en opy, ela i e en opy, and quan um ene gy condi ions).
Bu hese unica ions o en s ill ely on nume ous ex e nal laws and pa ame e s, o
example:
2
* G a i y de i es Eins ein equa ions h ough independen ly assumed Eins einHilbe
ac ion;
* Gauge eld heo y gi es he S anda d Model h ough he ex e nally added g oup
SU(3) ×SU(2) ×U(1)
and i s ep esen a ions;
* Mass spec a and coupling cons an s o ma e elds exis as expe imen al inpu s;
* Fluid and many-body eec i e equa ions (like Na ie S okes and Fokke Planck)
a e de i ed h ough independen app oxima ions.
In o he wo ds, e en unde highly unied s uc u al amewo ks, wha a e physical
laws and wha a e alues o de ailed pa ame e s s ill e ain massi e deg ees o eedom,
appea ing no as unique consequences o some highe -le el p inciple.
1.2 Limi Unica ion App oach wi hou Obse e On ology
Many unica ion schemes a emp o u he cons ain physical laws by in oducing ob-
se e s, compu a ion, o in o ma ion p ocessing on ologies (e.g., iewing he uni e se
as some obse a ioncompu a ion ne wo k). Howe e , such schemes o en s uggle o
main ain o mal objec i i y and may in oduce addi ional me aphysical assump ions.
This pape delibe a ely **in oduces no addi ional obse e on ology concep s**, bu
ea s obse e s only as a de i ed s uc u e wi hin he uni e se on ology objec (e.g.,
local ope a o subalgeb as and s a es on ce ain wo ldlines), and places he en i e bu den
o unica ion on he ollowing h ee ypes o no-obse e on ology in insic s uc u es:
1. **Unied Time Scale**: The scale mo he o mula
κ(ω)
gi en by sca e ing phase,
ela i e densi y o s a es, and g oup delay ace, se ing as he sole sou ce o all physical
ime eadings;
2. **Bounda y Time Geome y**: Composed o space ime bounda y, induced me ic,
second undamen al o m, and o al connec ion
Ω∂
, uni ying g a i y, gauge elds, and
esolu ion ow in o bounda y bundle geome y;
3. **Gene alized En opy and Causal S uc u e**: Dening gene alized en opy
Sgen
on small causal diamonds, and cons aining geome y and quan um s a es using i s ex-
emum and mono onici y.
On his basis, we cons uc a global consis ency unc ional
I[U]
and p opose he
unied consis ency p inciple
δI[U] = 0
. This pape will p o e: local and hie a chical
expansions o his p inciple na u ally yield all amilia physical laws.
1.3 Pape S uc u e
Sec ion 2 denes he Uni e se On ology Objec , unied ime scale, bounda y ime geom-
e y, and gene alized en opy s uc u e. Sec ion 3 cons uc s he consis ency unc ional
I[U]
and p esen s he unied consis ency p inciple. Sec ion 4 de i es Eins ein equa ions
and quan um ene gy condi ions a he le el o small causal diamonds. Sec ion 5 de i es
gauge equa ions and eld con en cons ain s a he le el o bounda y
K
- heo y and
o al connec ion. Sec ion 6 de i es local quan um eld heo y and Wa d iden i ies a
he quan umsca e ing le el. Sec ion 7 de i es uid dynamics and many-body eec i e
g adien ows in he coa se-g ained limi . Appendix p o ides echnical p oo s o key
o mulas and heo ems.
3
2 Uni e se On ology Objec and Unied S uc u e
2.1 Uni e se On ology Objec
We cha ac e ize he uni e se as an on ological ma hema ical objec
U=Ue , Ugeo, Umeas, UQFT, Usca , Umod, Uen ,
(7)
whe e: 1.
Ue = (M, g, ≺)
is a globally hype bolic Lo en zian mani old wi h causal
pa ial o de
≺
; 2.
Ugeo
con ains a amily o small causal diamonds
{Dp, }
, B own
Yo k quasilocal s ess enso , and GibbonsHawkingYo k bounda y e m; 3.
Umeas =
(A∂, ω∂)
is he bounda y obse able algeb a and s a e; 4.
UQFT = (Abulk, ωbulk)
is he bulk
quan um eld heo y algeb a and s a e; 5.
Usca = (S(ω;ℓ), Q(ω;ℓ))
is he equency
esolu ion dependen sca e ing ma ix and Wigne Smi h g oup delay ma ix; 6.
Umod
is he Tomi aTakesaki modula s uc u e and modula ow induced by
(A∂, ω∂)
; 7.
Uen
is gene alized en opy
Sgen
, ela i e en opy, and quan um ene gy condi ions on small
causal diamonds.
No e: No obse e on ology is in oduced in his deni ion; obse e s can be subse-
quen ly iewed as specic choices o local subalgeb as and s a es in
UQFT
on wo ldlines.
2.2 Unied Time Scale
In sca e ing heo y, le
S(ω;ℓ)
be he sca e ing ma ix a ene gy
ω
and esolu ion
ℓ
, i s
de e minan phase is
φ(ω;ℓ) = a g de S(ω;ℓ),
(8)
g oup delay ma ix
Q(ω;ℓ) = −iS(ω;ℓ)†∂ωS(ω;ℓ),
(9)
i s ace
Q(ω;ℓ)
and spec al shi unc ion de i a i e
ρ el(ω;ℓ)
sa is y he scale iden i y
unde na u al egula i y condi ions
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
(10)
We call
κ(ω)
he unied scale densi y, and dene unied ime scale
τ
(ane eedom
omi ed) ia
τ(E) = ZE
−∞
κ(ω) dω.
(11)
The unied ime scale axiom s ipula es: all physical ime eadings (p ope ime, edshi
ime, modula ow pa ame e , e c.) in he uni e se belong o he same scale class
[τ]
.
2.3 Bounda y Time Geome y and To al Connec ion
Conside bulk egion
MR⊂M
and i s bounda y
∂MR
. Dene induced me ic
hab
,
ou wa d no mal
na
, second undamen al o m
Kab
on
∂MR
, and GibbonsHawkingYo k
bounda y e m
SGHY =1
8πG Z∂MR
Kp|h|dd−1x.
(12)
4
All geome y and in e ac ions on he bounda y a e uniedly encoded in o he o al
connec ion
Ω∂=ωLC ⊕AYM ⊕Γ es,
(13)
whe e: 1.
ωLC
is Le iCi i a connec ion, cha ac e izing space ime g a i a ional geome-
y; 2.
AYM
is YangMills connec ion on in e nal gauge g oup, cha ac e izing elec oweak,
s ong in e ac ions, e c.; 3.
Γ es
is connec ion on esolu ion space, cha ac e izing eno -
maliza ion g oup ow and obse a ional esolu ion change.
I s cu a u e decomposes as
F(Ω∂) = R⊕FYM ⊕F es,
(14)
co esponding o cu a u e o space ime, gauge eld s eng h, and esolu ion ow.
2.4 Gene alized En opy and Causal S uc u e
On
(M, g, ≺)
, selec a poin
p
and pa ame e
, cons uc small causal diamond
Dp, =J+γ(− )∩J−γ( ),
(15)
whe e
γ(τ)
is a imelike geodesic passing h ough
p
. Dene gene alized en opy
Sgen(Dp, ) = A(∂Dp, )
4Gℏ+Sbulk(Dp, ),
(16)
whe e
Sbulk
is he on Neumann en opy o bulk quan um elds on
Dp,
.
The unica ion equi emen is: unde app op ia e cons ain s, he  s -o de a ia ion
ex emum condi ion o
Sgen
on small causal diamond amily
{Dp, }
yields g a i a ional
eld equa ions, second-o de a ia ion yields quan um ene gy condi ions and ocusing
condi ions, while he nes ed amily
{Dτ}
along unied scale pa ame e
τ
sa ises gene -
alized en opy mono onici y, he eby dening mac oscopic ime a ow.
3 Consis ency Func ional and Unied Consis ency P in-
ciple
3.1 S uc u e o Consis ency Func ional
Dene cosmic consis ency unc ional on he abo e s uc u e
I[U] = Ig a +Igauge +IQFT +Ihyd o,
(17)
co esponding o consis ency cons ain s a geome icen opy, gauge opological, quan um
sca e ing, and coa se-g ained le els espec i ely.
1. Geome icEn opy Te m
Ig a =1
16πG ZM
(R−2Λ)p|g|ddx+1
8πG Z∂M
Kp|h|dd−1x−λen X
D∈Dmic oSgen(D)−S∗
gen(D),
(18)
whe e
S∗
gen(D)
is en opy ex emum unde xed ex e nal condi ions,
Dmic o
is he amily
o small causal diamonds co e ing
M
.
5

2. GaugeTopological Te m
Igauge =Z∂M×Λh FYM ∧⋆FYM+µ op CS(AYM) + µKIndex(D[E])i,
(19)
whe e
[E]∈K(∂M ×Λ)
is he
K
-class o channel bundle,
CS
is Che nSimons e m,
Index(D[E])
is he index o Di ac ope a o coupled o
E
.
3. Quan umSca e ing Te m
IQFT =X
D∈Dmic o
SωD
bulk∥ωD
sca ,
(20)
whe e
ωD
bulk
is es ic ion o bulk s a e on
D
,
ωD
sca
is e e ence s a e p edic ed by sca e ing
da a and unied scale,
S(·∥·)
is ela i e en opy.
4. Coa se-G ained Fluid Te m
Ihyd o =ZMhζ(∇µuµ)2+η σµνσµν +X
k
Dk(∇µnk)2ip|g|ddx,
(21)
whe e
uµ
is mac oscopic eloci y eld,
σµν
shea enso ,
nk
densi ies o conse ed quan-
i ies, coecien s
ζ, η, Dk
de e mined by
Γ es
and mic oscopic sca e ing da a.
3.2 Unied Consis ency P inciple
**Unied Consis ency P inciple**
The Uni e se On ology Objec
U
mus sa is y: unde all allowed a ia ions
δgab, δE, δΩ∂, δωbulk, δ(Γ es, uµ, nk)
(22)
we ha e
δI[U]=0.
(23)
In o he wo ds, he eal uni e se is a s uc u e ha makes he consis ency unc ional
I[U]
achie e s able ex emum.
Subsequen sec ions will demons a e: Eule Lag ange condi ions o his unied con-
sis ency p inciple a die en le els a e p ecisely he physical laws we know.
4 Geome ic Le el: Small Causal Diamonds and Ein-
s ein Equa ions
4.1 Expansion o Small Causal Diamonds
In oduce Riemann no mal coo dina es in neighbo hood o
p∈M
, such ha
gab(p) = ηab, ∂cgab(p) = 0.
(24)
Take imelike uni ec o
ua
, le
γ(τ)
be geodesic sa is ying
γ(0) = p, ˙γ(0) = u
. Fo
sucien ly small
, causal diamond
Dp, =J+(γ(− )) ∩J−(γ( ))
(25)
has olume and bounda y a ea expansions
V(Dp, ) = αd dh1 + c1Rab(p)uaub 2+O( 3)i,
(26)
A(∂Dp, ) = βd d−1h1 + c2Rab(p)uaub 2+O( 3)i.
(27)
6
4.2 Fi s -O de Va ia ion o Gene alized En opy
Gene alized en opy is
Sgen(Dp, ) = A(∂Dp, )
4Gℏ+Sbulk(Dp, ).
(28)
Va ia ion wi h espec o me ic
δgab
gi es
δSbulk(Dp, ) = −1
2ZDp, p|g| ⟨Tab⟩δgab ddx.
(29)
Subs i u ing
δA(∂Dp, )
and
δSbulk
in o
δIg a ∼X
Dp, h1
4GℏδA(∂Dp, ) + δSbulk(Dp, )−λen δ(Sgen −S∗
gen)i.
(30)
In limi
→0
, equi ing
δIg a = 0
o any local
δgab
yields
Gab + Λgab = 8πG⟨Tab⟩.
(31)
Thus, Eins ein equa ions a e necessa y condi ions o unied consis ency p inciple a
geome icen opy le el, no independen axioms.
4.3 Second-O de Va ia ion and Quan um Ene gy Condi ions
Fu he conside ing de o ma ion along ligh ay di ec ions, analyzing second-o de a ia-
ion o
Sgen
, and u ilizing quan um in o ma ion inequali y
δ2Sgen ≥0
(32)
can de i e local o ms o quan um ene gy condi ions and quan um ocusing conjec u e.
This ensu es s abili y o geome icen opy s uc u e and consis ency o ime a ow unde
unied scale.
5 GaugeTopological Le el: Bounda y
K
-Class and Field
Con en Unica ion
5.1 Bounda y Channel Bundle and
K
-Class
On
∂M ×Λ
, equency esolu ion dependen sca e ing ma ix
S(ω;ℓ)
ac s on channel
space
Hchan(ω, ℓ)
a each
(ω, ℓ)
. These channel spaces glue in o be bundle
E→∂M ×Λ,
(33)
wi h s uc u e g oup es ic ed uni a y g oup
U es
. I s s able equi alence class
[E]∈K(∂M ×Λ)
(34)
uniedly encodes: * Gauge g oups and ep esen a ions (de e mined by s uc u e g oup
and associa ed bundle); * Fe mi/Bose s a is ics and chi ali y (de e mined by
Z2
g ading
and spin s uc u e); * Topological phases and p o ec ed bounda y modes (de e mined by
K
-class in a ian s).
7
5.2 Consis ency Va ia ion and YangMills Equa ions
Va ying
AYM
unde xed
[E]
, we ha e
δIgauge =Z∂M×Λ
δAYM ∧⋆∇µFµν
YMdd−1x+· · · ,
(35)
equi ing
δIgauge = 0
o any
δAYM
yields
∇µFµν
YM =Jν
YM,
(36)
i.e., YangMills equa ions wi h sou ce, whe e
Jν
YM
comes om momen um conse a ion
and bounda ybulk coupling.
Thus, gauge eld equa ions a e Eule Lag ange condi ions o unied consis ency p in-
ciple a gauge opological le el.
5.3 Index Cons ain and Field Con en Selec ion
The e is a na u al pai ing be ween index o Di ac ope a o
D[E]
coupled o
E
Index(D[E])∈Z
(37)
and sca e ing
K1
class
[S]∈K1(∂M ×Λ)
⟨[E],[S]⟩= Index(D[E]).
(38)
Index e m in consis ency unc ional
µKIndex(D[E])
(39)
equi es in a iance unde allowed a ia ions, imposing condi ions simila o anomaly can-
cella ion: only hose
[E]
a e allowed ha make index and sca e ing class pai ing sa is y
specic cons ain s.
In o he wo ds, which gauge g oup and eld con en he uni e se chooses is no an
ex e nal assump ion in his amewo k, bu a solu ion o
K
- heo y index and sca e ing
consis ency equa ions.
6 Quan umSca e ing Le el: Rela i e En opy Fixed
Poin and Local QFT
6.1 Rela i e En opy Consis ency
On each small causal diamond
D
, ac ual bulk s a e es ic ion is
ωD
bulk
, e e ence s a e
ωD
sca
is gi en by sca e ing da a
S(ω;ℓ)
, unied scale
κ(ω)
, and bounda y ime geome y
ia some sca e ing- o-s a e cons uc ion.
Consis ency e m
IQFT =X
D
SωD
bulk∥ωD
sca 
(40)
has a ia ional p ope ies: * Fi s -o de a ia ion is ze o a
ωD
bulk =ωD
sca
; * Second-o de
a ia ion gi es Fishe in o ma ion me ic, ensu ing s abili y o minimum.
Unied consis ency p inciple equi es
ωD
bulk =ωD
sca
o be a minimum o all
D
.
8
6.2 Wigh man Axioms and QFT Recons uc ion
Unde unied ime scale and causali y axioms,
n
-poin unc ions
Wn(x1, . . . , xn)
o sca e ing
scale e e ence s a e
ωD
sca
sa is y: 1. Lo en z co a iance and locali y; 2. Mic ocausali y
(commu a i i y o ope a o s a spacelike sepa a ion); 3. Spec um condi ion (ene gy
spec um has lowe bound); 4. Posi i i y and clus e decomposi ion.
By Wigh man econs uc ion heo em, one can cons uc Hilbe space
H
, acuum
s a e
Ω
, local algeb a amily
{A(O)}
, and uni a y ep esen a ion
U(a, Λ)
, he eby ob-
aining local quan um eld heo y
Q= (H,Ω,{A(O)}, U).
(41)
The e o e, local QFT is no an independen assump ion, bu an ine i able esul o
unied consis ency p inciple a quan umsca e ing le el.
6.3 Field Equa ions and Wa d Iden i ies
Fu he , a ying sca e ing ma ix
S(ω;ℓ)
and co esponding G een unc ions, equi ing
IQFT
s abili y unde xed unied scale and bounda y
K
class, leads o:
1. Exis ence o local eld ope a o s
ϕa(x)
sa is ying Eule Lag ange equa ions
Ea(ϕ, ∂ϕ, . . . ) = 0,
(42)
whose mass spec um and coupling cons an s a e uniquely de e mined by sca e ing an-
aly ic in a ian s;
2. In e nal symme ies co espond o Noe he cu en s
Jµ
, whose Wa d iden i ies a e
gi en by in a iance o
IQFT
unde symme y a ia ion;
3. LSZ limi and LehmannSymanzikZimme mann s uc u e ensu e equi alence be-
ween eld heo y and sca e ing desc ip ions.
Thus, eld equa ions and symme y s uc u es a e also unied back in o consis en
a ia ional p inciple.
7 Coa se-G ained Limi : Unica ion o Fluid Dynam-
ics and Many-Body G adien Flow
7.1 Resolu ion Connec ion and Mac oscopic Conse ed Cu en s
In long-wa eleng h and low- esolu ion limi s, esolu ion connec ion
Γ es
induces p ojec ion
om mic oscopic deg ees o eedom o mac oscopic conse ed quan i ies. Fo example,
o ene gymomen um and pa icle numbe conse a ion, he e a e mac oscopic cu en s
Tµν
hyd o, Jµ
a,
(43)
sa is ying
∇µTµν
hyd o = 0,∇µJµ
a= 0.
(44)
Consis ency unc ional
Ihyd o =ZMhζ(∇µuµ)2η σµνσµν +X
k
Dk(∇µnk)2ip|g|ddx
(45)
minimiza ion condi ion gi es iscosi y and diusion con olled mac oscopic dynamics,
i.e., gene alized Na ie S okes equa ions and diusion equa ions.
9