Final Unification of Physical Laws: Unique Solution of Consistent Variational Principle on Universe Ontological Object

Final Uni ica ion o Physical Laws: Unique Solu ion o
Consis en Va ia ional P inciple on Uni e se
On ological Objec
Haobo Ma1Wenlin Zhang2
1Independen Resea che
2Na ional Uni e si y o Singapo e
No embe 19, 2025
Abs ac
In p e ious wo k, we ha e cha ac e ized he “Uni e se” as a maximal, consis en ,
and comple e on ological objec Uwi hin a gi en ounda ional ca ego y, p o iding
uni ied ime scale, bounda y ime geome y, causal–en opic–obse e axioms, and
uni ied encoding o “de ail da a” D= ([E],{ α}). This sys em achie es uni ica-
ion a s uc u al and pa ame ic le els bu e ains a c i ical gap: physical laws
hemsel es s ill appea in an in insic, agmen ed manne , e.g., Eins ein
equa ions, Yang–Mills equa ions, Di ac equa ions, Na ie –S okes equa ions, mul i-
agen esou ce alloca ion dynamics, e c., ha e no ye been demons a ed as unique
consequences o a single on ological p inciple.
The goal o his pape is o b idge his gap: wi hin he amewo k o he Uni e se
On ological Objec U, uni ied ime scale, and bounda y ime geome y, we cons uc
asingle consis en a ia ional p inciple
δI[U] = 0,(1)
and p o e:
1. This a ia ional p inciple consis s only o h ee uni e sal equi emen s: (i)
Causal–Sca e ing Consis ency (uni a i y and mac oscopic causali y); (ii) Gene -
alized En opy Mono onici y and S abili y (“Gene alized Second Law” unde uni-
ied ime scale); (iii) Obse e –Consensus Consis ency (all local obse e s’ models–
eadou s mus be embeddable in o he same U).
2. On small causal diamonds, a ia ion o geome y and s a es de i es Eins ein
equa ions Gab + Λgab = 8πG⟨Tab⟩and ene gy–momen um conse a ion om his
p inciple;
3. Unde ixed geome y and uni ied scale, a ia ion o bounda y channel bun-
dles and ield con en de i es local gauge in a iance and Yang–Mills ield equa ions;
4. On gi en geome y and gauge s uc u e, a ia ion o ma e ields and sca -
e ing da a de i es Di ac/Klein–Go don ield equa ions and local quan um ield
heo y (sa is ying mic ocausali y and spec al condi ions);
5. In long-wa eleng h and coa se-g ained limi s, a ia ion o esolu ion con-
nec ions and en opy unc ionals de i es gene alized hyd odynamics and e ec i e
Na ie –S okes equa ions, as well as en opy g adien low dynamics o mul i-agen
esou ce alloca ion.
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Thus, gene al ela i i y, gauge ield heo y, quan um ield heo y, luid
and s a is ical physics, and mul i-agen dynamics a e all shown o be
necessa y condi ions o he same uni e se consis en a ia ional p inci-
ple a di e en esolu ion le els and bounda y condi ions. This s ic ly
comple es physical uni ica ion: he e a e no longe mu ually independen “ o ce
laws” o “ma e equa ions”, bu only one Uni e se On ological Objec Uand one
consis en a ia ional p inciple. Speci ic heo ies a e me ely e ec i e un oldings o
his p inciple in di e en limi s.
The appendices p o ide speci ic cons uc ion o his consis ency unc ional I[U],
a ia ional de i a ion on small causal diamonds, uni ied ea men o gauge s uc-
u e and ield con en , and ou lines o econs uc ion p oo s o local quan um ield
heo y and hyd odynamic limi s.
Keywo ds: Uni e se On ological Objec ; Consis en Va ia ional P inciple; Final Uni i-
ca ion; Uni ied Time Scale; Gene alized En opy; Eins ein Equa ions; Yang–Mills; QFT;
Hyd odynamics
1 In oduc ion
1.1 P oblem Re o mula ion
In he adi ional pic u e, he main h ead o “uni ied physics” un olds oughly along wo
pa hs:
1. **S uc u al Uni ica ion**: Placing space- ime, causali y, en opy, obse e , and
o he basic concep s wi hin a single geome ic–in o ma ion amewo k, such as causal
mani olds, axioma ic QFT, holog aphic duali y, and in o ma ion geome y.
2. **De ail Uni ica ion**: W i ing “pa ame e s and s uc u es” o di e en ields
(high ene gy, condensed ma e , cosmology, mul i-agen , e c.) as he same ma hema ical
da a, such as K- heo y classes o gauge g oups and ep esen a ions, analy ic in a ian s
o sca e ing, e c.
The p e ious Uni e se On ological Objec Uand uni ied de ail da a D= ([E],{ α})
achie ed hese wo s eps: all physical sys ems (including high-ene gy sca e ing, opo-
logical phases, cosmological backg ounds, and mul i-agen ne wo ks) can be iewed as
subs uc u es o U, hei “de ails” uni o mly encoded as bounda y K-classes and sca e -
ing analy ic in a ian s. Howe e , om a physics pe spec i e, his s ill does no cons i u e
“ inal uni ica ion”:
* Eins ein equa ions a e s ill sepa a ely assumed as “geome ic laws”; * Yang–Mills
and Di ac equa ions a e s ill sepa a ely assumed as “ ield heo y laws”; * Na ie –S okes
equa ions and Fokke –Planck equa ions a e s ill sepa a ely in oduced as “mac oscopic
laws”; * Resou ce–s a egy dynamics o mul i-agen sys ems a e s ill sepa a ely modeled
as some op imiza ion o game p ocess.
In o he wo ds, **we ha e uni ied he “iden i ies o s age and ac o s”, bu ha e no
ye uni ied he “sole sou ce o he sc ip ”**.
1.2 Co e P oposi ion o This Pape
This pape p oposes: wi hin he Uni e se On ological Objec U, he e exis s a single,
consis en a ia ional p inciple
δI[U]=0,(2)
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which elies only on he ollowing minimal, physically non-nego iable equi emen s:
1. **Causal–Sca e ing Consis ency**: The sca e ing p ocess o any small causal
diamond mus be embeddable in o a single global uni a y e olu ion wi hou iola ing
mac oscopic causali y; 2. **Gene alized En opy Mono onici y and S abili y**: Un-
de uni ied ime scale, he gene alized en opy unc ional Sgen o small causal diamonds
mus sa is y app op ia e mono onici y and ex emum s abili y unde cons ain s o a oid
uncon olled nega i e ene gy and in o ma ion pa adoxes; 3. **Obse e –Consensus Con-
sis ency**: Models and eadou s o any ini e obse e ne wo k mus be embeddable in o
he same uni e se s a e U, and consensus can be eached ia uni ied scale and causal
s uc u e.
We will p o e:
* W i ing hese h ee equi emen s as a single “Uni e se Consis ency Func ional” I[U]
and a ying all a iable deg ees o eedom (geome y, channel bundles, connec ions, ield
con en , s a es, esolu ion lows, and obse e models), he ob ained “Eule –Lag ange
condi ions” a e espec i ely equi alen a di e en le els o: * G a i a ional ield equa-
ions on small causal diamonds (GR); * Gauge ield equa ions on bounda y channel
bundles and o al connec ions (Yang–Mills); * Local wa e equa ions and gauge Wa d
iden i ies on bulk ields (QFT); * Hyd odynamics, di usion, and mul i-agen en opy
g adien low in coa se-g ained limi s.
Thus, in his sense, “physical laws” a e uni ied as **un oldings o a single uni e se
consis en a ia ional p inciple a di e en le els**.
1.3 Pape S uc u e
Sec ion 2 e iews he co e s uc u es o Uni e se On ological Objec U, uni ied ime
scale, and bounda y ime geome y. Sec ion 3 gi es he speci ic cons uc ion o Uni e se
Consis ency Func ional I[U]. Sec ion 4 pe o ms a ia ion o geome y and s a es on small
causal diamonds o de i e Eins ein equa ions. Sec ion 5 pe o ms a ia ion o bounda y
K-classes and o al connec ions unde ixed geome y o de i e gauge ield equa ions and
ield con en cons ain s. Sec ion 6 pe o ms a ia ion o ma e ields and sca e ing da a
unde gi en geome y and gauge backg ound o de i e local quan um ield heo y and
Wa d iden i ies. Sec ion 7 de i es hyd odynamics and mul i-agen en opy g adien low
in coa se-g ained limi s. Appendices p o ide comple e p oo ou lines o main heo ems.
2 Uni e se On ological Objec and Uni ied S uc u e
Re iew
2.1 Uni e se On ological Objec
The Uni e se On ological Objec is w i en as
U=Ue , Ugeo, Umeas, UQFT, Usca , Umod, Uen , Uobs, Uca , Ucomp,(3)
whe e:
1. Ue = (M, g, ≺) is globally hype bolic Lo en zian mani old wi h causal pa ial
o de ; 2. Ugeo con ains amily o small causal diamonds {Dp, }, Gibbons–Hawking–Yo k
bounda y e ms, and B own–Yo k quasi-local s ess enso s; 3. Umeas = (A∂, ω∂) is
bounda y obse able algeb a and s a e; 4. UQFT = (Abulk, ωbulk) is bulk quan um ield
3
heo y; 5. Usca = (S(ω;ℓ), Q(ω;ℓ)) is esolu ion– equency decomposed sca e ing ma ix
and Wigne –Smi h g oup delay; 6. Umod is Tomi a–Takesaki modula low induced by
(A∂, ω∂); 7. Uen con ains gene alized en opy Sgen on small causal diamonds, ela i e
en opy, and quan um ene gy condi ions; 8. Uobs is amily o obse e s and consensus
geome ies {Oi}; 9. Uca gi es ca ego y o mo phisms be ween abo e s uc u es; 10.
Ucomp desc ibes compu a ion and complexi y bounda ies in he uni e se.
2.2 Uni ied Time Scale and Scale Iden i y
Uni ied ime scale is gi en by sca e ing scale mo he o mula:
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω),(4)
whe e φ(ω) = a g de S(ω) is sca e ing phase, ρ el(ω) spec al shi unc ion de i a i e
( ela i e densi y o s a es), Q(ω) = −iS†(ω)∂ωS(ω) g oup delay ma ix.
Uni ied ime scale axiom s ipula es: all physical ime eadou s in he uni e se a e
a ine ans o ma ions o he same scale class [τ].
2.3 Bounda y Time Geome y and To al Connec ion
On bounda y ∂MR, de ine o al connec ion
Ω∂=ωLC ⊕AYM ⊕Γ es,(5)
whose cu a u e is
F(Ω∂) = R⊕FYM ⊕F es,(6)
co esponding o space ime cu a u e, gauge ield s eng h, and cu a u e o esolu ion
low espec i ely.
2.4 Gene alized En opy and Obse e Consensus Geome y
Fo small causal diamond Dp, , gene alized en opy is de ined as
Sgen(Dp, ) = A(∂Dp, )
4Gℏ+Sbulk(Dp, ),(7)
whe e Sbulk is on Neumann en opy o bulk quan um ields.
Obse e Oiis o malized as
Oi=Ci,≺i,Λi,Ai, ωi,Mi, Ui, ui,{Cij},(8)
wi h local causal domain, esolu ion hie a chy, obse able algeb a and s a e, model amily
and upda e ope a o , and communica ion s uc u e.
P io wo k shows: unde uni ied ime scale and ela i e en opy mono onici y con-
di ions, global consensus causal s uc u e and uni ied ime a ow can be cons uc ed on
obse e ne wo k.
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3 Cons uc ion o Uni e se Consis ency Func ional
I[U]
This sec ion cons uc s he co e objec o his pape : Uni e se Consis ency Func ional
I[U].
3.1 Th ee Classes o Consis ency Requi emen s
We decompose “physical ealizabili y o uni e se” in o h ee classes o consis ency e-
qui emen s:
1. **Causal–Sca e ing Consis ency**: Sca e ing ma ix amily S(ω;ℓ) de ined on
any ini e bulk egion MRand i s bounda y ∂MRmus be ex ensible o a global uni a y
e olu ion, and co esponding G een’s unc ions suppo obeys causal cone s uc u e.
2. **Gene alized En opy Mono onici y and S abili y**: Fo any nes ed amily o
small causal diamonds {Dτ}on a imelike cu e, gene alized en opy Sgen(Dτ) sa is ies
app op ia e mono onici y and second-o de a ia ional s abili y condi ions unde uni ied
ime scale pa ame e τ, a oiding uncon olled nega i e ene gy and en opy dec ease.
3. **Obse e –Consensus Consis ency**: Models and eadou s o any ini e obse e
ne wo k {Oi}mus be embeddable in o he same Us a e unde uni ied scale and causal
pa ial o de , and consensus can be eached ia communica ion and upda es wi hou
leading o essen ial con adic o y desc ip ions o he same physical p ocess.
3.2 Fo m o Consis ency Func ional
We w i e consis ency equi emen s as a unc ional
I[U] = Ig a [g, ωbulk]+Igauge[E, Ω∂]+IQFT[Abulk, ωbulk]+Ihyd o[Γ es, Sgen]+Iobs[{Oi}],(9)
and claim:
* Causal–Sca e ing Consis ency mainly cons ains IQFT and Igauge; * Gene alized
En opy Mono onici y and S abili y mainly cons ains Ig a and Ihyd o; * Obse e –
Consensus Consis ency mainly cons ains Iobs and IQFT.
Speci ic o ms a e as ollows.
3.2.1 G a i y–En opy Te m
Le
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),
(10)
whe e S∗
gen(D) is en opy ex emum unde gi en ex e nal condi ions, λen >0 is a La-
g ange mul iplie , Dmic o is a amily o small causal diamonds co e ing M. The las e m
penalizes con igu a ions de ia ing om en opy ex emum.
3.2.2 Gauge–Geome ic Te m
On bounda y, consis ency equi emen s o channel bundle E→∂M ×Λ and o al
connec ion Ω∂a e w i en as
Igauge =Z∂M×Λ (FYM ∧⋆FYM) + µ op ·CS(AYM) + µK·Index(D[E]),(11)
5

whe e CS is Che n–Simons e m, Index(D[E]) is index o Di ac ope a o on K-class [E].
This e m ensu es consis ency o gauge s uc u e wi h K-class, and penalizes con igu a-
ions iola ing gauge Wa d iden i ies.
3.2.3 QFT–Sca e ing Te m
Consis ency o bulk QFT is gi en by a ela i e en opy ype unc ional
IQFT =X
D∈Dmic o
SωD
bulk∥ωD
sca ,(12)
whe e ωD
bulk is es ic ion o ac ual s a e on causal diamond D,ωD
sca is “ e e ence s a e”
p edic ed by sca e ing da a and uni ied scale, S(·∥·) is ela i e en opy. This e m
equi es local QFT model o be compa ible wi h sca e ing–scale p edic ions.
3.2.4 Fluid–Resolu ion Te m
In coa se-g ained limi , esolu ion connec ion Γ es, mac oscopic low ield uµ, and con-
se ed cu en s Jµsa is y a amily o en opy p oduc ion inequali ies. We w i e
Ihyd o =ZMζ(∇µuµ)2+η σµνσµν +X
k
Dk(∇µnk)2p|g|ddx, (13)
whe e σµν is shea enso , nkconse ed quan i y densi ies, ζ, η, Dk iscosi y and di u-
sion coe icien s de e mined by Γ es. This e m equi es mac oscopic e olu ion o ollow
p inciple o minimum en opy p oduc ion.
3.2.5 Obse e –Consensus Te m
Consis ency o obse e ne wo k is w i en as
Iobs =X
i
Sωi∥ωbulk|Ci+X
(i,j)
SCij∗(ωi)∥ωj,(14)
whe e i s e m penalizes de ia ion o obse e ’s in e nal model om ue uni e se s a e
on i s causal domain, second e m penalizes inconsis ency be ween models a e commu-
nica ion.
3.3 Consis en Va ia ional P inciple
**Uni ied Consis ency P inciple**
Unde p emises o uni ied ime scale and causal–en opic–obse e axioms, physical
uni e se co esponds o Uni e se On ological Objec Usuch ha
δI[U] = 0 (15)
holds o all allowed a ia ions (including a ia ions o g, E, Ω∂, ωbulk,{Oi}).
In ollowing sec ions, we discuss physical meaning o his a ia ional condi ion laye
by laye .
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4 Geome ic Va ia ion on Small Causal Diamonds
and G a i a ional Field Equa ions
4.1 Va ia ional Se up
Conside a imelike geodesic γ(τ) and a amily o small causal diamonds Dp, nea i ,
whe e p=γ(0), ≪Lcu . We a y me ic gab inside Dp, and bulk s a e ωbulk, keeping:
1. Ex e nal geome y and ex e nal s a e ixed; 2. Uni ied ime scale κ(ω) una ec ed
o i s o de ; 3. Gene alized en opy cons ain s implemen ed ia Ig a +IQFT.
4.2 Fi s -O de Va ia ion o Gene alized En opy and Eins ein
Equa ions
Fo i s -o de a ia ion o Sgen(Dp, ), unde ixed olume o app op ia e cons ain s,
δSgen =1
4GℏδA(∂Dp, ) + δSbulk.(16)
Geome ic pa δA can be expanded as local unc ion o cu a u e Rab a p, bulk en opy
a ia ion δSbulk exp essed ia s ess–ene gy enso ⟨Tab⟩. Subs i u ing in o condi ion
δIg a = 0, in limi →0, de i es
Gab + Λgab = 8πG⟨Tab⟩,(17)
i.e., Eins ein equa ions. De ailed de i a ion in Appendix A.
4.3 Second-O de Va ia ion and Quan um Ene gy Condi ions
Fo second-o de a ia ion o same small causal diamond, conside ing de o ma ion di-
ec ion along some ligh ays, second-o de a ia ion o gene alized en opy ela es o
quan um in o ma ion inequali ies, ob aining quan um ene gy condi ions and ocusing
condi ions. These condi ions ensu e s abili y o g a i a ional backg ound and unidi ec-
ionali y o mac oscopic ime a ow.
5 Va ia ion o Bounda y Channel Bundles and To al
Connec ion: Uni ica ion o Gauge Fields and Field
Con en
5.1 Channel Bundle K-Class and Gauge S uc u e
On ixed geome ic backg ound, a y bounda y channel bundle Eand o al connec ion
Ω∂, equi ing:
1. K-class [E] o channel bundle ixed (only s able equi alence a ia ions allowed); 2.
Compa ibili y o K1class o sca e ing ma ix S(ω;ℓ) wi h [E] main ained.
In Igauge, a ia ion o AYM gi es
∇µFµν
YM =Jν
YM,(18)
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i.e., Yang–Mills equa ions, whe e sou ce Jν
YM comes om coupling o bounda y and bulk
s a es. Allowed a ia ions o [E] and ex emum condi ion o Di ac index e m cons ain
ield con en and chi al s uc u e: only ield con en s ensu ing anomaly cancella ion and
ze o index pai ing a e allowed. De ailed a gumen in Appendix B.
5.2 Physical Meaning o Uni ied Gauge S uc u e
Abo e esul s imply:
* “Which gauge ields, which ma e ields, coupled in wha ep esen a ions” a e no
longe ex e nal “inpu s”, bu esul s o ex emum condi ions o Igauge; * Gauge in a iance
and Wa d iden i ies o igina e om in a iance o Iunde a ia ion o Ω∂; * “Field con en ”
and “gauge g oup” a e exp essions o channel bundle K-class and o al connec ion, no
independen en i ies.
6 Va ia ion o Bulk Fields and Sca e ing Da a: Uni-
ica ion o Local QFT
6.1 De i a ion o Local Field Equa ions om Rela i e En opy
Func ional
Unde gi en geome y and gauge backg ound, a y bulk QFT s a e ωbulk and ope a o
s uc u e Abulk. Rela i e en opy unc ional
IQFT =X
D
SωD
bulk∥ωD
sca (19)
equi es ac ual s a e o be as close as possible o e e ence s a e p edic ed by sca e ing–
scale on each small causal diamond. Va ying ωbulk yields a amily o “local consis ency
condi ions”, sa is ying:
1. Mic ocausali y: obse ables a spacelike sepa a ed poin s commu e; 2. Spec al
Condi ion: ene gy spec um bounded below unde uni ied ime scale; 3. Dynamics: ield
ope a o s sa is y se o local wa e equa ions (e.g., Klein–Go don, Di ac), whose mass
spec um and couplings a e de e mined by analy ic in a ian s in D.
De ailed de i a ion elies on Wigh man unc ion econs uc ion and a ia ional p op-
e ies o ela i e en opy, see Appendix C.
6.2 Wa d Iden i ies and Sca e ing–Field Theo y Compa ibili y
Va ying sca e ing ma ix S(ω;ℓ) i sel , unde ixed uni ied scale, channel bundle K-class,
and uni a i y, equi ing non-inc ease o IQFT, can de i e Wa d iden i ies and LSZ limi
condi ions, ensu ing consis ency be ween ield heo y and sca e ing desc ip ions.
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7 Coa se-G ained Limi : Hyd odynamics and Mul i-
Agen En opy G adien Flow
7.1 Resolu ion Connec ion and Mac oscopic Fluids
In long-wa eleng h and low- esolu ion limi s, we ca e no abou all deg ees o eedom o
mic oscopic ields, bu ini e se o conse ed cu en s Jµ
aand mac oscopic eloci y ield
uµ. Resolu ion connec ion Γ es gi es p ojec ion om mic oscopic deg ees o eedom o
mac oscopic a iables and low ules on mani old.
Va ying Γ es and mac oscopic a iables, minimiza ion condi ion o en opy p oduc ion
unc ional
Ihyd o =ZMζ(∇µuµ)2+η σµνσµν +X
k
Dk(∇µnk)2p|g|ddx(20)
gi es gene alized Na ie –S okes equa ions and di usion equa ions:
∇µTµν
hyd o = 0,∇µJµ
a= 0,(21)
whe e s ess enso Tµν
hyd o and cu en Jµ
acon ain iscosi y and di usion e ms.
7.2 En opy G adien Flow o Mul i-Agen Sys ems
Viewing mul i-agen sys em as obse e ne wo k {Oi}, whose s a egy dis ibu ions and
belie s a es e ol e wi h uni ied scale. Obse e –consensus unc ional
Iobs =X
i
Sωi∥ωbulk|Ci+X
(i,j)
SCij∗(ωi)∥ωj(22)
a ying ωigi es a amily o g adien low equa ions, simila o na u al g adien descen
o mi o descen , en opy unc ional dec eases mono onically unde uni ied scale.
In con inuous limi , such g adien lows sha e same s uc u e as mac oscopic hyd o-
dynamics: bo h can be w i en as g adien low o some gene alized en opy S
∂τρ=−g adGS(ρ),(23)
whe e Gis me ic de e mined by causal–geome ic and esou ce cons ain s.
8 S ic Meaning o Physical Uni ica ion
In abo e cons uc ion, “uni ica ion” no longe means “exis ence o a la ge symme y
g oup” o “exis ence o an all-encompassing Lag angian”, bu means:
1. **On ological Uni ica ion**: Only one Uni e se On ological Objec U, con ain-
ing geome y, channel bundles, connec ions, ields, en opy, and obse e s; 2. **Scale
Uni ica ion**: All imes and scales uni ied by scale mo he o mula κ(ω); 3. **Va ia-
ional Uni ica ion**: Exis ence o a single Uni e se Consis ency Func ional I[U], whose
ex emum condi ions a e equi alen a di e en le els o GR, gauge ield heo y, QFT,
hyd odynamics, mul i-agen en opy g adien low, e c.; 4. **De ail Uni ica ion**: All
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