Energy–Flow Cosmology v1.2: Foundational Framework and Cross-Field Continuity

Ene gy–Flow Cosmology 1.2: Founda ional F amewo k and C oss-Field Con inui y
Mo en Magnusson
Ene gy–Flow Cosmology Ini ia i e, No way
(Da ed: No embe 7, 2025)
Ene gy–Flow Cosmology (EFC) is o mula ed as a co a ian , non-equilib ium he modynamic
ield amewo k ex ending Gene al Rela i i y (GR). A scala ene gy- low po en ial
E
go e ns
en opy-d i en o ganiza ion and cu a u e e ec s. In he equilib ium limi , EFC educes o GR;
ou o equilib ium, i ep oduces key la ge-scale phenomena wi hou explici da k-ma e o da k-
ene gy pos ula es. The amewo k is ma hema ically well-posed, admi s a Lyapuno unc ional
consis en wi h he second law, and is designed o c oss-da ase Bayesian es ing wi h a pa simonious,
hie a chically scaled pa ame e se . Beyond cosmology, he same ield law maps o in o ma ional and
biological sys ems ia an explici unc ional co espondence o a ia ional ee ene gy, sugges ing a
single he modynamic subs a e ac oss six as ophysical classes and six in e disciplina y domains.
This 1.2 manusc ip p o ides he ounda ional de i a ion and con inui y o he 2.1 (uni ied
amewo k) and 2.2 (applied c oss- ield in eg a ion) p ep in s.
I. INTRODUCTION
Mode n cosmology explains obse a ions h ough GR
plus he phenomenological componen s o ΛCDM. While
empi ically powe ul, his s anda d model posi s da k
ma e and da k ene gy wi hou di ec mic ophysical
iden i ica ion. He e we ad ance Ene gy–Flow Cosmology
(EFC): a he modynamic ield amewo k in which en opy
g adien s d i e ene gy lows ha mani es as cu a u e,
s uc u e o ma ion, and la e- ime expansion.
EFC es s on h ee claims: (i) a single co a ian ield
law o non-equilib ium ene gy/en opy low unde lies
g a i a ional phenomena; (ii) he law is ma hema ically
well-posed and he modynamically consis en ; (iii) he
same unc ional o m ex ends o in o ma ional/biological
domains h ough an ene gy–en opy co espondence. Ou
objec i e in 1.2 is no o eplace GR bu o gene alize i
o non-equilib ium and o p o ide a compac , alsi iable
o malism wi h anspa en pa ame e economy.
A. Six as ophysical classes (EFC-S)
The amewo k a ge s a uni ied accoun o : (i) galaxy
o a ion cu es, (ii) ea ly massi e galaxies (JWST), (iii)
cosmic expansion, (i ) CMB low-
ℓ
elaxa ion, ( ) cosmic
oids, ( i) g a i a ional lensing. EFC aims o i hese
wi h a single ield law and a sha ed global pa ame e se .
B. Six in e disciplina y domains (EFC-C)
The same law applies concep ually o: (i) biology
(me abolic o ganiza ion), (ii) ecology (ene gy- a e densi y
and complexi y), (iii) neu oscience (en opy landscapes
o b ain s a es), (i ) in o ma ion heo y (Landaue link),
( ) economics ( esou ce/ low cons ain s), ( i) machine
lea ning/AI ( ee-ene gy minimiza ion). Sec ion VI gi es
he unc ional b idge o a ia ional ee ene gy.
No a ion and con en ions. Signa u e (
−,
+
,
+
,
+),
c
=
ℏ
= 1, and
M−2
P
= 8
πG
. Ma e densi y
ρ
; co a ian
de i a i e
∇µ
. Spa ial domain Ω
⊂R3
wi h ou wa d
no mal
n
. Local en opy
S
=
S
(
E , x
). We de ine he
en opic d i e
σ
(
E , S
)
≡∂S/∂V
and he educed sou ce
F(E , S).
II. MATHEMATICAL FORMULATION
A. Ac ion, ield equa ion, and dis o mal coupling
On (M, gµν ), conside he conse a i e ac ion
S=Zd4x√−gM2
P
2R+1
2K(S)gµν ∂µE ∂νE −V(E , S)+Sm[ψ, ˜gµν ],
(1)
wi h
K
(
S
)
>
0 an en opic coupling and
V
(
E , S
) a
po en ial encoding he en opic d i e. Ma e ields
ψ
couple o a dis o mal me ic
˜gµν =A(E , S)gµν +B(E , S)∂µE ∂νE .(2)
Va ia ion w. . . E yields
∇µ
K(S)∇µE −∂V
∂E
=J[E , S;ψ],(3)
whe e
J
collec s sou ce e ms induced by
(2)
. We link
explici ly o en opy by choosing
∂V
∂E
=λ σ(E , S), σ ≡∂S
∂V ,(4)
wi h a dimensional cons an
λ
. In he quasi-s a ic, weak-
ield limi , abso bing
J
in o
F
, we eco e he ellip ic
co e
−∇·K(S)∇E =F(E , S), F ≡λ σ −J.(5)
This gene alizes Poisson (
K→K0
,
F∝ρ
) and admi s
MOND-like p-Laplace egimes when K∝ |∇E |p−2.
2
S ess–ene gy and conse a ion. The scala con ibu-
ion is
T(E )
µν =K(S)∂µE ∂νE −gµν 1
2K(S)∂αE ∂αE −V(E , S),
(6)
and di eomo phism in a iance implies
∇µTµν
(m)
+
Tµν
(E )= 0.
B. Causali y and he modynamic a ow
(GENERIC/Onsage )
I e e sibili y is in oduced ia he Rayleigh dissipa ion
unc ional (ou side he conse a i e ac ion),
R=1
2Zd4x√−g τ (∂ E )2,(7)
which in GENERIC/Onsage yields he Maxwell–
Ca aneo o m
τ ∂ E −∇·K(S)∇E =λ σ(E , S)−J.(8)
Time- e e sal symme y is b oken by
R
, en o cing he
he modynamic a ow.
C. GR limi and FLRW backg ound
Va ying (1) w. . . gµν gi es
M2
PGµν =T(m)
µν +T(E )
µν .(9)
In equilib ium (
σ→
0,
K→K0
),
E
is e ec i ely s a ic
and GR is eco e ed. Fo a homogeneous backg ound
(FLRW), olume-a e aging (3) implies
3H2≃8πG ρb+ Λe ( ),Λe ( ) = ⟨λ σ⟩
K0
.(10)
D. Well-posedness and Lyapuno s abili y
Theo em A (exis ence/uniqueness, s a ic). Assume
Kmin >
0 and Lipschi z con inui y o
K
(
S
) and
F
(
E , S
) in
E
. Wi h Di ichle /Neumann da a
on
∂
Ω, Eq.
(5)
admi s a unique weak solu ion
E ∈H1
(Ω) by mono one-ope a o me hods (coe ci -
i y/hemicon inui y) and Schaude ixed poin (compac -
ness ia Rellich–Kond acho ).
Theo em B (Lyapuno /second law). De ine
E[E ] = ZΩ1
2K(S)|∇E |2−V(E , S)d3x. (11)
Unde
(8)
and mild egula i y,
dE/d ≤
0; equi alen ly
dS/d ≥0.
Dimensional no e. Choosing [
E
] so ha
[K]|∇E |2∼[F] ensu es scala consis ency o (5).
TABLE I. Illus a i e join compa ison (placeholde s): EFC
s. ΛCDM wi h c oss-p edic ion.
Da ase NΛCDM NEFC AIC(ΛCDM) AIC(EFC) ln BF
SPARC (calib a ion) 1 1 X1Y1ln BF1
CFHTLenS (holdou ) 1 0 X2Y2ln BF2
SNe-Ia (holdou ) 1 1 X3Y3ln BF3
BAO (holdou ) 1 0 X4Y4ln BF4
III. SCALING LAW AND PARAMETER
HIERARCHY
Pe ec scale in a iance ac oss domains is nei he eal-
is ic no equi ed. We adop a minimal eno malisa ion-
s yle scaling o he global pa ame e ec o Θ:
Θ(µ) = Θ0+Alog µ
µ0
,(12)
wi h
µ
a cha ac e is ic scale (galaxy, clus e , cosmic)
and A(2–3 sha ed coe icien s) ixed ac oss as ophysical
da ase s. Func ional amilies a e es ic ed o
K(S)=K0(1+βS)p, F(E , S)=a0ρ+a1ργ,(13)
wi h Θ = {K0, β, p, a0, a1, γ}sha ing (12).
IV. BAYESIAN VALIDATION AND NO-GO
TESTS
Model selec ion uses AIC, BIC, and log Bayes ac o s
(
ln BF
) on independen da ase s
Di
: SPARC o a ion
cu es, CFHTLenS weak lensing, SNe-Ia (p ogeni o -age
co ec ed), BAO. Calib a ion is pe o med on SPARC;
Θ(
µ
) is hen held ixed o CFHTLenS/SNe/BAO ( ue
c oss-p edic ion). P io s a e weakly-in o ma i e wi h
ini e suppo ; we epo pos e io iden i iabili y and
pos e io -p edic i e checks.
P e- egis e ed no-go es s.
•
NG-1 (as o): Θ(
µ
) calib a ed on SPARC ails (
>
3σ) on CFHTLenS.
•
NG-2 ( he mo): p edic ed halo empe a u e
Th
(
)
disag ees wi h X- ay/SZ a ixed ρ( ).
•
NG-3 (in o): unc ional isomo phy o a ia ional
ee ene gy (Sec. VI) b eaks due o non-con exi y.
V. ENTROPIC HALO TEMPERATURE: A
UNIQUE EFC SIGNATURE
We de ine he En opic Halo Tempe a u e as a di ec ly
es able he modynamic obse able:
kBTh( )≡1
n( )
1
2K(S)|∇E |2,(14)
3
Ene gy–Flow Cosmology (EFC) – Single The modynamic Field Law
Galaxy o a ion Ea ly galaxies Expansion
CMB low-ℓVoids Lensing
Biology Ecology Neu oscience
In o ma ion Economics AI/ML
FIG. 1. Concep ual mapping: six as ophysical classes ( op)
and six in e disciplina y domains (bo om) go e ned by he
same non-equilib ium ield law.
wi h pa icle numbe densi y
n
(
). F om
(5)
and
(13)
,
Th
(
) is p edic ed om ba yonic
ρ
(
) wi hou in oking
pa icle DM. The p o ile can be compa ed agains X-
ay b igh ness and Sunyae –Zel’do ich (SZ) measu e-
men s, o e ing an EFC-speci ic disc imina o ela i e
o EG/MOND.
VI. INFORMATIONAL EQUIVALENCE:
EFC–FEP BRIDGE
Le he EFC unc ional be
FEFC[E ] = Z1
2K(S)|∇E |2−V(E , S)dx. (15)
Va ia ional ee ene gy in he F ee Ene gy P inciple (FEP)
eads
F a (q) = Eq[−log p(x, z)] −H(q),(16)
(accu acy minus complexi y). Iden i y he ene ge ic and
en opic pa s unde a small-noise in o ma ion-geome y
app oxima ion:
Eq[−log p]↔1
2K(S)|∇E |2, H(q)↔V(E , S)/λ,
using
(4)
. Unde mild con exi y/coe ci i y, minimiza ion
o
FEFC
induces he same descen di ec ion as
F a
. Thus
EFC does no claim o eplace FEP; i induces he same
a ia ional s uc u e om a he modynamic ield law.
(Fo mal lemmas omi ed he e; o be p o ided in a echnical
supplemen .)
VII. SIX+SIX SCHEMATIC (VISUAL)
VIII. CROSS-FIELD INTEGRATION AND
CONTINUITY
This 1.2 manusc ip p o ides he ounda ional, co a i-
an o mula ion, scaling law, and alsi ica ion p o ocol.
Two complemen a y p ep in s ex end his wo k:
•
EFC 2.1 — Uni ied The modynamic F ame-
wo k ac oss S uc u e, Dynamics, and Cog-
ni ion
DOI: 10.6084/m9. igsha e.30478916.
Expands he heo e ical a chi ec u e he e in o a
sys em-le el schema linking EFC-S/D/C and do-
main on ologies.
•
Applied EFC 2.2 — C oss-Field In eg a ion
Summa y (2025)
DOI: 10.6084/m9. igsha e.30530156.
Ope a ionalizes Eqs.
(5)
–
(8)
wi h he scaling law
(12)
and epo s p elimina y Bayesian c oss- ield i s
and applica ion-le el me ics.
Toge he , 1.2
→
2.1
→
2.2 de ine a con inuous p og am:
law ( his pape ), amewo k mapping ( 2.1), and applied
in eg abili y ( 2.2).
IX. DISCUSSION AND OUTLOOK
EFC in eg a es he modynamic i e e sibili y wi h co-
a ian dynamics, yielding a single ield law ha (i)
educes o GR a equilib ium, (ii) is ma hema ically
well-posed and Lyapuno s able, and (iii) suppo s pa -
simonious, c oss-da ase Bayesian alida ion unde a
minimal scaling hie a chy. The En opic Halo Tem-
pe a u e p o ides a dis inc i e obse a ional signa u e;
he EFC–FEP b idge cla i ies c oss- ield ele ance wi h-
ou o e claiming. Immedia e p io i ies include join i s
(SPARC
→
CFHTLenS/SNe/BAO wi h ixed Θ(
µ
)), quan-
i a i e
Th
(
) p edic ions e sus X- ay/SZ, and a echnical
appendix de i ing mic o- o-meso closu es o
K
(
S
) and
λ.
ACKNOWLEDGMENTS
The au ho hanks colleagues and e iewe s o con-
s uc i e discussions. Any e o s a e he au ho ’s own.
4
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