1
Disc e e Cosmology
Model: Rela i is ic
G oup Delays as a
Tes able O igin o
G a i y and Redshi
Nick Ma ko
Bulga ian Academy o Sciences
[email p o ec ed]
Abs ac
The Disc e e Cosmology Model (DCM)
e o mula es g a i a ion and cosmic
dynamics by in oducing a causal desc ip ion
o mass. E e y mass is ea ed as a ini e-
sized domain whose in e nal in e ac ions
p opaga e a ini e speed. Because he o ces
ha main ain a body’s shape and olume a e
no ins an aneous, ela i is ic delays occu
wi hin he mass i sel . In his amewo k,
ma e co-expands wi h space, and he local
expansion speed is in insically ela i is ic
a he han es ic ed o he low- eloci y
Hubble a e. The in e nal ime lags dis o he
equilib ium geome y: egions expe iencing
sho e in e ac ion delays expand in o mo e-
delayed egions, gene a ing a ne expansion-
delay g adien ha mani es s
mac oscopically as g a i y. The same
p inciple applies ac oss scales— om
quan ized pa icles o galaxies—p o iding a
uni ied explana ion o g a i a ional
a ac ion, galac ic o a ion, and
cosmological edshi o e ing an al e na i e
o da k-ma e –dependen desc ip ions by
a ibu ing he same e ec s o ini e-speed,
delay-d i en mechanics. A he pa icle le el,
expansion and o a ion (spin) canno occu
simul aneously a he ela i is ic limi ; hei
disc e e al e na ion p oduces quan ized delay
s eps ha de ine he ene gy s a es o ma e .
On la ge scales, he cumula i e g adien s o
hese ini e-speed in e ac ions go e n he
cu a u e and appa en accele a ion o he
Uni e se. A co e DCM p edic ion, equa ing
elec omagne ic and g a i a ional dila ions, is
empi ically suppo ed by seismic wa e
eloci ies on Ea h, Ma s, and he Moon,
which align wi h escape eloci ies. By
linking mass geome y, ini e p opaga ion
speed, and delay-d i en scaling, he DCM
o e s a es able causal amewo k o
g a i a ion and cosmology.
Keywo ds: ini e-speed in e ac ions; g oup
delay; disc e e expansion; quan ized mass;
seismic alida ion; g a i y wi hou da k
ma e ; cosmological edshi .
1 In oduc ion
Con en ional cosmology ea s mass as an
abs ac scala sou ce o space ime cu a u e
and g a i a ional po en ial [1,2]. In he
Disc e e Cosmology Model (DCM) [3], mass
is ins ead desc ibed as a ini e domain
sus ained by in e nal in e ac ions ha
p opaga e a ini e speed. Because he
main enance o a body’s shape is no
ins an aneous, ela i is ic delays occu
be ween i s inne and ou e egions. These
small bu cumula i e ime o se s dis o local
equilib ium and c ea e g adien s o expansion
delay, p oducing an appa en a ac ion
be ween adjacen masses. A less-delayed
2
mass e ec i ely expands in o he di ec ion o
a mo e-delayed one, es ablishing he causal
basis o g a i a ional beha io . The same
ini e-speed p inciple ope a es
hie a chically— om pa icles o galaxies—
o e ing an al e na i e o da k-ma e –
dependen desc ip ions by a ibu ing he
same e ec s o ini e-speed, delay-d i en
mechanics. A mic oscopic scales, pa icle
o a ion and expansion al e na e disc e ely,
making he in e ac ions quan ized; a
cosmological scales, o e lapping delay
g adien s go e n he obse ed galac ic
o a ion and edshi . This pe spec i e
e ames g a i a ion and cosmological
expansion as eme gen consequences o
ini e-speed olume main enance, uni ying
phenomena adi ionally a ibu ed o da k
ma e and da k ene gy wi hin a single delay-
mechanics amewo k. The amewo k is
suppo ed by empi ical co ela ions, such as
he obse ed iden i y be ween seismic wa e
eloci ies and plane a y escape eloci ies on
Ea h, Ma s, and he Moon, which con i m
he sha ed dila ion limi p edic ed by he
model. Unlike he s anda d cosmological
iew whe e ma e emains s a ic wi hin an
expanding me ic, he DCM pos ula es ha
mass i sel co-expands wi h space and ha
his in insic expansion p oceeds a
ela i is ic g oup speeds limi ed by local
delay mechanics.
Th oughou he his o y o g a i a ion heo y,
a ac ion has been modeled ei he
geome ically, as cu a u e o space ime in
gene al ela i i y [1,2], o
phenomenologically, as a o ce media ed by
in isible componen s such as da k ma e and
da k ene gy [4–7]. Bo h app oaches
ep oduce obse a ions bu equi e pos ula es
ha emain physically un e i ied:
ins an aneous cu a u e esponse in he i s
case and unde ec ed mass–ene gy in he
second. DCM eplaces hese assump ions
wi h explici ime-delay mechanics. I ea s
g a i a ional and cosmological e ec s as
mani es a ions o ini e-speed sel -
in e ac ion wi hin ex ended masses. The
ela i is ic delay be ween an inne and an
ou e egion o a mass elemen gene a es an
e ec i e g adien o expansion and
con ac ion, p oducing he same o bi al and
edshi beha io a ibu ed o ex e nal ields
o unseen ma e . Unlike modi ica ions o
New onian o ela i is ic dynamics [8–10],
his amewo k does no al e undamen al
equa ions o mo ion bu ede ines he sou ce
e ms as delayed-in e ac ion domains. In
doing so, i p ese es ela i is ic causali y
while o e ing a uni ied explana ion o he
phenomena con en ionally asc ibed o da k
ma e and da k ene gy.
The ollowing sec ions o malize his delay-
mechanics amewo k by quan i ying how
disc e e expansion s eps, ela i is ic
o a ional coupling, and ini e p opaga ion
speed combine o ep oduce g a i a ional
a ac ion, galac ic o a ion, and
cosmological edshi wi hin a single causal
model.
2 Physical P inciples
2.1 Disc e e Expansion Lag
While specula i e, his sec ion in oduces a
disc e e-expansion hypo hesis p o iding he
minimal kinema ic mechanism equi ed o
adial g oup delays in he nex sec ions. The
3
empi ical p edic ions de eloped la e
(seismic–escape con e gence, o a ion
cu es, and edshi supp ession) depend
only on he exis ence o cumula i e delays,
no on he mic ophysical de ails o §2.1.
Quan um mechanics does no in e p e spin
as li e al pa icle o a ion, since accoun ing
o he measu ed magne ic momen his way
would equi e supe luminal su ace speeds—
assuming a ixed mass and adius. Special
ela i i y, howe e , allows o mass inc ease
a ela i is ic speeds, which al e s he
dynamics o o a ion and angula momen um.
Figu e 1: S ep expansion o a pa icle, showing
o e shoo /unde shoo (a schema ic illus a ion o he
hypo hesis).
The Disc e e Cosmology Model (DCM)
builds on his by p oposing ha pa icle mass
a ies disc e ely a Comp on equencies,
suppo ing ela i is ic su ace mo ion and
disc e e adial g ow h. This oscilla o y mass
beha io econciles he obse ed magne ic
momen wi h ela i is ic limi s and p o ides
a de e minis ic physical mechanism o
in insic spin.
DCM hypo hesizes ha pa icles
o e shoo /unde shoo space’s expansion due
o disc e e ela i is ic delays caused by hei
o a ion, p oducing s epwise g ow h (Fig. 1).
Spin, ein e p e ed as a iable-mass o a ion,
sepa a es expansion (in e ac ion-hea y) and
o a ion (minimal expansion and in e ac ion,
high magne ic momen ) phases a Comp on
equencies (Fig. 2). A phase-weigh ed oy
model o he gy omagne ic a io, based on
a iable ine ia a Comp on equency, is
de eloped in Appendix F o he
Supplemen al Ma e ial.
Figu e 2: Rein e p e a ion o pa icle spin as a iable-
mass o a ion: la ge size indica ing lowe mass. Pa icle
size seen om he expanding space pe spec i e (Fig. 1). A
schema ic illus a ion o he hypo hesis
Mo eo e , his amewo k o e s a new
pe spec i e on quan um unneling: i spin
a ises om ela i is ically o a ing a iable
mass, hen mass-ene gy could ansien ly
exceed classical h esholds, allowing
pa icles o bypass ene gy o momen um
ba ie s in a manne consis en wi h unneling
obse a ions.
This in e p e a ion may ha e expe imen al
implica ions, pa icula ly in sys ems
in ol ing spin-pola ized unneling,
anomalous magne ic esponses, o ime-
esol ed sca e ing a Comp on-scale
in e als. Mo eo e , hea ie lep ons may be
unde s ood as esonan o e shoo s a es ha
4
occu when elec on’s disc e e delays
accumula e cohe en ly (Appendix H).
The nex sec ions will ocus on g a i y whe e
empi ical co ela ions o e subs an ial
e idence suppo ing DCM. Al hough he
p esen wo k ocuses on g a i a ional and
cosmological scales, whe e cumula i e g oup
delays domina e, he same p inciple may
mani es a he quan um scale as disc e e
delays, consis en wi h he oscilla o s in
hyd odynamic quan um analogs [11-13]; a
de ailed ea men o spec al s uc u e,
howe e , lies beyond he scope o his pape .
2.2 Expansion Delay and G oup
Phenomena
The hypo hesis o es is ha g a i y a ises
om cumula i e ela i is ic delays in masses,
whe e inne pa icles mus pe pe ually
displace ou e laye s, a ini e in e ac ion
speeds. He e, g a i y is a collec i e e ec ,
no sou ced by single pa icles.
The objec i e would be o p o e ha he
hypo hesized g oup delay ma ches he
g a i a ional ime dila ion om he
Schwa zschild me ic:
𝒕𝟎=𝒕𝒇 √𝟏−𝒗𝒆𝒔𝒄
𝟐(𝒓)
𝒄𝟐 (1)
exp essed as a unc ion o he escape eloci y
𝑣𝑒𝑠𝑐. The GR-consis en o mula ion o he
delay-based s ess–ene gy enso is p esen ed
in Appendix A, whe e i s decomposi ion,
closu e, and conse a ion p ope ies a e
de i ed. Al hough in oduced heu is ically,
he delay-based me ic used in he nex
sec ions e ec i ely encodes g a i a ional and
kinema ic ime dila ion, and in Appendix A i
is shown o be consis en wi h Eins ein’s ield
equa ions.
2.3 Seismic–G a i a ional Veloci y
Con e gence as Empi ical E idence
o Rela i is ic G oup Delay
A di ec empi ical es o he Disc e e
Cosmology Model (DCM) is p o ided by he
nume ical con e gence be ween seismic
p opaga ion speeds wi hin plane a y in e io s
and he escape eloci ies o he plane s.
Rela i is ic ime dila ions can be associa ed
wi h bo h speeds: kinema ic and
g a i a ional. I g a i y a ises om
cumula i e ini e-speed delays o in e nal
in e ac ions, hese wo eloci ies should
coincide wi hin unce ain y o cohe en ly
s uc u ed bodies.
A adial P-wa e will be es ed as a
mac oscopic analogue o he in e nal
cohesi e p opaga ion ha main ains a
plane ’s shape and olume. As a i s check,
P-wa es ake app oxima ely 16 o 20 minu es
o c oss he Ea h's diame e . Which
immedia ely pu s he plane ’s escape eloci y
o 11.2 𝑘𝑚 𝑠⁻¹ in he ange o he a e age
adial P-wa e eloci ies. The bes empi ical
ma ch, howe e , was ound o occu when
using he a e age P-wa e speed in he
plane a y co e as a p oxy. The eason why he
co e alues a e mos ep esen a i e will be
discussed in Appendix A.8. Fo he Ea h’s
co e, he mean global P-wa e eloci y (≈
11.2 𝑘𝑚 𝑠⁻¹) equals i s escape eloci y
(11.2 𝑘𝑚 𝑠⁻¹) [16]. Fo Ma s, InSigh
obse a ions gi e 5.0 km s⁻¹, again ma ching
(𝒗𝒆𝒇𝒇 = 5.0 𝑘𝑚 𝑠⁻¹ [17]. Fo Venus, in e io
5
modeling yields 10.3 ± 0.4 𝑘𝑚 𝑠⁻¹ agains
10.4 km s⁻¹ [18].
Tidally locked bodies equi e a
ci cum e en ial p oxy because adial
p opaga ion is pa ially cons ained. In such
cases, he e ec i e g oup-delay eloci y
ollows he geome ic a io
𝒗𝒆𝒇𝒇 ≃𝒗𝑷−𝒘𝒂𝒗𝒆
𝜋, ep esen ing he ansi ion
om adial o ci cum e en ial cohe ence. Fo
he Moon, his gi es (𝒗𝒆𝒇𝒇 =𝟕.𝟒
𝜋=
2.4 km s⁻¹), equal o he luna escape
eloci y (2.4 km s⁻¹) [19]. He e, 7.4 km/s is
he a e age adial P-wa e speed ob ained by
in eg a ing he seismic wa e p opaga ion
du a ions along he adius om cen e o
su ace.
In each example, g a i a ional and
mechanical dila ions con e ge nume ically:
𝒗𝒆𝒇𝒇/𝒗𝒆𝒔𝒄 ≃1, (2)
which co esponds o
𝒗𝒆𝒔𝒄 ≃ 𝒗𝑷−𝒘𝒂𝒗𝒆
𝒓𝒂𝒅 (3)
implying ha he ini e-speed cohe ence
main aining he body’s olume ope a es a
he same ela i is ic limi ha de ines i s
g a i a ional po en ial. He e, 𝒗𝑷−𝒘𝒂𝒗𝒆
𝒓𝒂𝒅
ep esen s he adial componen o he mean
P-wa e speed.
Table 1 and Fig. 3 summa ize ep esen a i e
alues o Ea h, Ma s, Moon, Venus, and
Sun, whe e as e isks ma k modeled a he
han di ec ly measu ed da a.
Recen helioseismic in e sions (BiSON,
GONG, HMI, and Pa ke -P obe–cons ained
2024–2025 models) yield a cen al acous ic
speed o ≈540 ± 8 km/s, abou 12–13% below
he Sun’s su ace escape eloci y o 618
km/s. Wi hin DCM his small esidual is
in e p e ed as he e ec o nuclea ene gy
injec ion in he sola co e, which sligh ly
accele a es he e ec i e g oup-p opaga ion
beyond he pu e delay limi . This
in e p e a ion is consis en wi h he obse ed
slow sola -wind asymme y, which o igina es
p edominan ly om egions o s onge
magne ic supp ession a he han he ho es
co onal holes.
Ac oss all di e en ia ed bodies, he a io
(𝑹=𝒗𝒆𝒇𝒇/𝒗𝒆𝒔𝒄) emains nea uni y wi hin
unce ain ies, whe eas i egula as e oids all
well below his cohe ence limi .
This ela ion is no p edic ed by s anda d
plane a y-s uc u e models, which ea
seismic and g a i a ional pa ame e s as
independen .
Table 1: Empi ical da a suppo ing he eloci y con e gence law
backed by Apollo and InSigh missions o Moon and Ma s [14-
15].
Body
𝒗𝑷−𝒘𝒂𝒗𝒆
𝒓𝒂𝒅
km/s
𝒗𝒆𝒔𝒄
km/s
Ra io
𝑣𝑠/𝑣𝑒𝑠𝑐
Re e ence
Ea h
11.2
11.2
1.00
[16]
Ma s
5.0
5.0
1.00
[17]
Venus
10.3
10.4
1.00
[18]
Moon
7.4/π
2.4
1.00
[19]
As e oids
< 0.5
< 0.1
≫ 1
(diso de ed)
Es ima ed*
Sun
540
618
0.88
[20, 21]
* Based on in e io modeling
6
Figu e 3: Seismic–escape eloci y a io R wi h sou ce-based 1σ
unce ain ies. P oxies: Ea h—inne -co e P-wa e speed (PREM);
Ma s—co e P-wa e a CMB (InSigh ); Moon — man le P-wa e
speed /π (as in e ed om Apollo and GRAIL da a); Venus*—
Pe ple_X model sui e; Sun*—helioseismic sound speed (deep
in e io ). As e isks (*) indica e model/in e sion-based p oxies
a he han di ec co e seismology. π- a io applied o idal
locking.
In con as , DCM an icipa es such
con e gence na u ally: bo h seismic
ansmission and g a i a ional cu a u e
eme ge om he same ini e-speed delay
ield ha s abilizes he mass agains collapse.
When ha ield eaches ela i is ic
sa u a ion, i s e ec i e “delay modulus”
links seismic eloci y o escape eloci y
(Appendix A.5).
In e p e a ion wi hin he DCM
amewo k
In DCM, mass is de ined dynamically: as a
delay in he local expansion o disc e e
space ime elemen s due o ela i is ic
coupling wi h nea by mass. G a i y eme ges
as a mac oscopic consequence o his g oup
delay, and i s cumula i e e ec mani es s in
he o m o an escape- eloci y-quan i ied
space ime cu a u e. Simul aneously, he
abili y o a medium o ansmi in e nal
s esses (measu ed as seismic wa e speed) is
cons ained by he same delay mechanism,
namely, he p opaga ion ime o in e ac ions
ac oss he body's in e io .
Thus, he obse ed con e gence be ween and
signals a ela i is ic limi on in e nal signal
cohe ence. This sugges s ha seismic and
g a i a ional me ics a e no independen , bu
bo h eme ge om he same delay-go e ned
s uc u e o ma e .
Choice o wa e ype: Fo idally- ee bodies
P-wa e speeds a e used as he adial
in e ac ion p oxy; o idally locked bodies
we use P-wa e speeds as a ci cum e en ial
p oxy. This is a DCM hypo hesis and a di ec
es : i should be suppo ed by aniso opy
pa e ns; we do no assume i p o en. The
seismic-wa e a e age speeds and anges in
Fig. 3 a e aken om he e e ences. Escape
eloci y: 𝒗𝒆𝒔𝒄 =√2𝐺𝑀 𝑅
⁄ wi h mode n GM
and mean adius R; unce ain ies a e small s
seismic ones.
We ea he seismic–escape con e gence as
an empi ical egula i y p edic ed by DCM’s
g oup-delay mechanism. I is no assumed as
p oo o he mechanism; a he , i cons i u es
a alsi iable signa u e: g a i y-shaped co es
should sa is y once unce ain ies a e
p opaga ed. We p e-speci y he p oxy choice
and p o ide a p ospec i e a ge lis ;
de ia ions ou side he s a ed band would
alsi y his claim. Appendix A.8 illus a es
ha he same cohe ence–in e e ence pa e n
ecu s ac oss scales, culmina ing in he
Hubble ela ion.
2.4 Cosmological Redshi as
Expansion Delay
While he seismic co ela ion p o ides a
compelling local e i ica ion o he model's
ein e p e a ion o g a i y, he same
p inciples can be ex ended o cosmological
7
scales, whe e he cumula i e e ec o
disc e e delays mani es s as edshi .
Fo sys ems o g ouped masses, g a i y can
be analyzed om wo complemen a y
obse a ional pe spec i es: ha o an inside
wi hin he g a i a ional sys em, and ha o an
ou side obse ing om a dis an , non-
ine ial ame. D awing on he ele a o
analogy, he la -space ime inside
expe iences a longi udinal Dopple e ec ,
consis en wi h local ee- all condi ions. In
con as , he dis an obse e a he " op"
pe cei es a consis en wi h Eq. 1 ans e se
Dopple e ec , e lec ing ime dila ion ac oss
he g a i a ional ield.
In he s anda d cosmological model, he
edshi o ligh om dis an galaxies is
a ibu ed o he s e ching o space i sel —a
Dopple -like e ec due o me ic expansion.
Wi hin he Disc e e Cosmology Model, he
cosmological edshi is ein e p e ed as a
cumula i e g a i a ional ime delay
expe ienced by pho ons a e sing an
expanding acuum.
Unlike i ed-ligh hypo heses [22] ha in oke
pa h-leng h pho on a igue, DCM explains
edshi as an obse e - ela i e ime-dila ion
e ec om cumula i e in e ac ion delays,
he eby p ese ing image cohe ence [23] and
supe no a ime dila ion [24] while
simul aneously cons aining local
seismology and galac ic dynamics wi hin a
single, es able amewo k.
Figu e 4: Longi udinal Dopple and he obse e - ela i e
E en Ho izon
This delay is obse e - ela i e: he a he we
look, he mo e delayed he expansion o
ma e appea s o us. Ligh emi ed om such
egions o igina es om a slowe -clock
domain ela i e o he obse e ’s ame,
esul ing in a lowe obse ed equency, i.e.,
a edshi . Impo an ly, his edshi eme ges
wi hou he need o ecessional eloci y o
expanding me ic. I is he g a i a ional
analog o he longi udinal Dopple e ec
(Fig. 4) seen by a la -space ime obse e
looking in o Eins ein’s s a iona y
g a i a ional ele a o : he ele a o need no
mo e, ye he obse e pe cei es a edshi
due o ime dila ion.
This ein e p e a ion also p o ides a new
de i a ion o he Hubble law: cosmological
edshi esul s om g a i a ional delays, no
me ic expansion, scaling wi h dis ance R:
𝑣(𝑅) ~ √𝜌𝑅 (4)
de i ed om he escape eloci y o mula
ew i en in densi y (𝜌) e ms:
𝑣=√2𝐺𝑀 𝑅
⁄=√8 3
⁄𝜋𝐺𝜌 𝑅 (5)
8
Table 2: Densi y s. cosmic mean
Scale
Densi y s.
Mean
E idence
<10 Mpc
O e dense
2MASS,
SDSS
~50 Mpc
Possibly
o e dense
Laniakea
100–300
Mpc
Con lic ing
Mixed
claims
>300 Mpc
Cosmic mean
Planck CMB
The g a i a ional delay ac s as i he uni e se
is expanding in appea ance, bu no in
space ime i sel , dis inguishing DCM om
i ed ligh o ea lie non-me ic models.
Acco ding o Eq. 4, 18% local o e densi y
may explain he Hubble ension [25] o 8% –
9%. Table 2 poin s o po en ial sou ces o
o e densi y ha may a ec he ela ionship in
Eq. 4.
2.5 Redshi as cumula i e
g a i a ional delay: a minimal
de i a ion
We model he obse able edshi as a ising
om cumula i e ime dila ion along he
pho on pa h h ough an in e ac ion-limi ed,
disc e ely expanding medium. In he weak-
ield, s a iona y limi we use an e ec i e
iso opic me ic
ds2=−e2Φ𝑒𝑓𝑓
𝑐2𝑐2𝑑𝑡2 (6)
+e−2Φ𝑒𝑓𝑓
𝑐2(𝑑𝑟2+𝑟2dΩ2),
wi h he pa h-a e aged po en ial go e ning
clock a es o he medium. Fo null geodesics
he equency shi be ween emission a and
obse a ion a 0 is, o leading o de ,
1+𝑧≃𝑒𝑥𝑝(Φ𝑒𝑓𝑓(0)−Φ𝑒𝑓𝑓(𝑟)
𝑐2) (7)
≃1+Φ𝑒𝑓𝑓(0)−Φ𝑒𝑓𝑓(𝑟)
𝑐2
We decompose Φ𝑒𝑓𝑓 = Φ𝑔+ Φ𝑘 in o
(i) a g a i a ional delay e m
Φ𝑔 de e mined by he mass
dis ibu ion along he line o sigh
and
(ii) a kinema ic delay e m
Φ𝑘 accoun ing o he ini e-
speed suppo o expanding mul i-
body sys ems (see §2.7).
Fo cosmological sigh lines we app oxima e
Φ𝑔 by a slowly a ying unc ion o p ope
dis ance and expand o quad a ic o de in
/REH (REH an e ec i e e en -ho izon
scale, Fig. 7):
𝑧(𝑟)≃(𝐻0
𝑐)𝑟(1−𝑘 𝑟
𝑅𝐸𝐻), 0 ≤ ≲ REH,
(8)
whe e 𝑘 is a dimensionless coe icien
agg ega ing he cumula i e delay ela i e o
he linea Hubble law. This o m is
dimensionally consis en , educes o
Hubble’s law a small , and yields a
supp ession Δz/zlin ≃ 𝑘 a ≃ REH. Fi s o
p esen SN Ia+BAO econs uc ions sugges
𝑘 ≈ 0.08–0.10 i he en i e ension is
a ibu ed o delay.
The cosmological closu e o he delay enso
leading o his quad a ic edshi supp ession
is gi en in Appendix A.4.
9
2.6 In e p e ing 𝒌 om he line-o -
sigh po en ial
Le he line-o -sigh e ec i e po en ial be
Φ𝑒𝑓𝑓(𝑟)=1
𝑐∫𝑎∥(𝑠)𝑑𝑠
𝑟
0, whe e 𝑎∥ encodes
he e a ded in e ac ion coupling. In he
weak- ield limi he ac ional equency shi
accumula es as
𝑧(𝑟)≃1
𝑐2∫𝜕Φ𝑒𝑓𝑓(𝑠)
𝜕𝑠 𝑑𝑠
𝑟
0=1
𝑐2Φ𝑒𝑓𝑓(𝑟)
(9)
Assuming a smoo hly sa u a ing po en ial
Φ𝑒𝑓𝑓(𝑟)≃A −𝐵𝑟2
𝑅𝐸𝐻 (10)
wi h A≃𝐻0𝑐 and B≃k𝐻0𝑐 , we eco e he
quad a ic pa ame e iza ion abo e. The single
dimensionless pa ame e 𝑘 is he ( escaled)
a io o he ho izon-scale con ibu ion o he
linea Hubble e m. In da a applica ions can
be in e ed by a one-pa ame e eg ession o
H(z) o DL(z) agains ΛCDM baselines.
2.7 The CMB as Ho izon-Shell Re-
Emission
In he DCM amewo k, adia ion o igina ing
om beyond he obse able ho izon
expe iences cumula i e g oup-delay
sa u a ion a z
∗
≈ 1100. Using he longi udinal
Dopple ela ion 1+z=
√(1+β) (1−β)
⁄, his co esponds o an
e ec i e p opaga ion eloci y
β
∗
=0.99999835011, e =β
∗
c≈299,791.963 k
m/s, only ≈ 0.5 𝑘𝑚/𝑠 below he speed o
ligh .
This ini e delay limi ma ks he o ma ion o
a hin isibili y shell whe e ene gy is
sca e ed and e-emi ed wi h nea -Planck
spec al weigh ing. The obse ed
empe a u e ollows 𝑇obs =𝑇emi (1+𝑧∗
⁄),
yielding Tobs = 2.73 K o Temi ≈ 3000 K.
The shell’s nea -sphe ical geome y explains
he iso opy o he CMB, while small
aniso opies (𝛿𝑇/𝑇≈10−5) a ise om
inhomogenei ies in he ou e uni e se
p ojec ed on o he ho izon sc een.
This eplaces he need o a ho ea ly epoch
while p ese ing he CMB spec um and
pola iza ion ea u es, consis en wi h a
s a iona y-cu a u e uni e se.
The ho izon-shell mechanism is
ma hema ically equi alen o a sudden las -
sca e ing su ace in an o he wise s a iona y
uni e se, and he e o e inhe i s all success ul
ISW, SZ, and g a i a ional-lensing
p edic ions o ΛCDM o a ecombina ion-
like shell a z ≈ 1100.
The CMB’s obse ed E-mode pola iza ion
ampli ude (~5µK) and s ong E/B asymme y
can be in e p e ed in DCM as e-emission
om he same ho izon-shell. The shell has a
hin adial hickness (Δ
≲
0.1–1 Mpc;
iducially Δ ≈0.1 Mpc
≃
100 kpc),
which p ese es spec al pu i y and limi s
line-o -sigh damping.
The angula scale o he i s pola iza ion
peak ins ead e lec s he ans e se cohe ence
on he shell, wi h cha ac e is ic L
⊥∼
50- 100
Mpc, gi ing ℓpeak ≈𝜋 𝑅EH
𝐿⊥≈150–300, in
ag eemen wi h he i s E-mode peak
measu ed by Planck.
This in e p e a ion na u ally explains he
obse ed supp ession o p imo dial B-modes
and links CMB aniso opy o in e ac ions
16
undamen ally di e en in e p e a ion. In
ΛCDM, he accele a ion a ises om a
cosmological cons an Λ p oducing me ic
expansion; in DCM, i eme ges om causally
delayed disc e e expansion e en s o ma e
wi hin la space ime, main aining ene gy
conse a ion wi hou in oking Λ o da k
ene gy. A he e en -ho izon limi , bo h
amewo ks p edic simila magni udes, bu
DCM a ibu es he cu a u e en i ely o
g oup-delay s uc u e a he han o acuum
ene gy.
Hence, DCM p o ides a uni ying amewo k
in which:
• 𝑎DCM ep oduces MOND’s 𝑎0 a e
geome ic p ojec ion.
• he RAR cu e eme ges na u ally
om cumula i e g oup-delay
con ibu ions.
• and he ΛCDM ho izon accele a ion
appea s as a bounda y condi ion o
he same mechanism.
The quan i a i e ag eemen o DCM wi h
RAR cu a u e ac oss bo h galac ic and
clus e egimes, using a single geome ic
cons an 𝐻0, es ablishes a alsi iable baseline
o he nex -scale p edic ions discussed in
Sec ion 6.
6 Po en ial Expe imen al
Tes s
6.1 Tes ing Space ime Cu a u e
Tes 1: Measu e he ci cula P-wa e
p opaga ion on he Moon and he nea - adial
P-wa e p opaga ion on Ea h. Falsi ied by no
co ela ion.
6.2 Tes ing Galac ic Ro a ion
Tes 2: Analyze spi al galaxy and clus e
o a ion cu es (ALMA, spec oscopy) o
kinema ic ime dila ion. P edic s la cu es
wi hou da k ma e ; alsi ied by
inconsis ency.
6.3 Tes ing Cosmological Redshi
Tes 3: Measu e Local G oup edshi s
(spec oscopy, Cepheids). P edic s
g a i a ional delay; alsi ied by s anda d
Hubble law.
6.4 Quan um Sa u a ion and Pho on
Abso p ion
A he mic oscopic le el, he same sa u a ion
condi ion ex ends o pa icle in e ac ions.
Each elemen a y pa icle expands disc e ely
a i s Comp on equency 𝜈𝐶=𝑚𝑐2/ℎ, wi h
a local expansion on p opaga ing a 𝑐. A
pho on in e ac s when i s oscilla ing ield
becomes phase-cohe en wi h his Comp on
on , elimina ing g oup-delay misma ch:
Δ𝜏𝛾𝑒 →0. (26)
This ze o-delay condi ion allows he
pho on’s oscilla ion o me ge in o he
pa icle’s expansion cycle, p oducing
abso p ion. The cohe ence limi
𝑣Comp on =𝑐=𝑣esc (27)
hus, uni es pho on abso p ion and
g a i a ional apping as mani es a ions o
pe ec delay synch oniza ion. The cohe ence
17
is es able ia ime- esol ed pho on
abso p ion spec a nea Comp on
wa eleng hs (e.g., ens o MeV), as ou lined
in Supplemen C.
The logic he e is consis en wi h a iable-
mass dynamics in §2.1 (spin ein e p e a ion)
and Appendix H (lep on shells).
Supplemen C: Quan um Sa u a ion
Mechanism
A he quan um scale, DCM in e p e s pho on
abso p ion and emission as ansien
cohe ence phenomena wi hin disc e e
expansion shells. A pa icle’s Comp on
expansion on p opaga es a 𝑐, de ining a
mic oscopic ho izon o synch oniza ion.
When a pho on ield becomes phase-ma ched
o his on , he ela i e g oup delay
anishes,
Δ𝜏𝛾𝑒 →0, (28)
and he pho on’s ene gy me ges in o he local
delay po en ial Ψ. He e, Ψ ep esen s he
pa icle’s cumula i e delay ield ene gy,
modula ed by Comp on cycles.
The esul ing condi ion
𝑣Comp on =𝑐=𝑣esc, (29)
ep esen s quan um sa u a ion— he same
cohe ence limi ha de ines he mac oscopic
g a i a ional ho izon. Bo h p ocesses
co espond o ze o ela i e g oup delay,
whe e p opaga ion and expansion become
indis inguishable. Pho ons a e emi ed when
he local on o e shoo s equilib ium,
abso bed when i e-aligns, and e lec ed
when cohe ence canno be es ablished. This
amewo k uni ies op ical and g a i a ional
in e ac ions unde he same delay- a iance
p inciple.
The mic ophysical implica ions o DCM—
ex ending he delay-mechanics amewo k o
lep on magne ic momen s and Comp on-
scale sca e ing—a e discussed in
Appendix~H.5.
6.5 CMB Pola iza ion
The Disc e e Cosmology Model p edic s ha
he ain pola iza ion o he cosmic
mic owa e backg ound a ises no om
p imo dial ecombina ion, bu om
aniso opic sca e ing a a hin isibili y shell
nea he e en ho izon. A hin adial window
(Δ ≲ 0.1–1 Mpc; we adop Δ ≈ 0.1 Mpc ≃
100 kpc as a iducial alue) p ese es spec al
pu i y and limi s line-o -sigh damping, while
he angula scale o he pola iza ion peaks is
se by he ans e se cohe ence on he shell
(cha ac e is ic size L⊥ ∼ 50–100 Mpc),
yielding ℓ ≈ πREH/L⊥ ≈ 150–300, consis en
wi h Planck. This mechanism na u ally gi es
an E-mode ampli ude o ~5–10 μK, wi h
negligible p imo dial B-modes (lensing only)
and a apid decline o E–B c oss-powe
owa d la ge scales. The measu ed E-mode
ampli ude and i s angula dependence hus
p o ide a di ec es o he ho izon-sca e ing
in e p e a ion.
Disc e e Resonance In e p e a ion o he
CMB
The ΛCDM in e p e a ion o he CMB powe
spec um achie es an imp essi e nume ical
i by adjus ing a mul i-pa ame e amewo k
in ol ing ba yon densi y, cold da k ma e ,
cu a u e, eioniza ion, spec al il , and da k
ene gy. While success ul empi ically, his
18
app oach is essen ially a pos - ac o syn hesis
o esonan ha monics whose physical o igin
emains dis ibu ed among se e al
hypo he ical componen s. The esul ing
model ep oduces he obse ed spec um
h ough a complex pa ame e coupling a he
han h ough a single causal mechanism,
leading o wha may be desc ibed as a
s a is ical econs uc ion a he han a
physical explana ion.
In con as , he Disc e e Cosmology Model
(DCM) de i es he same ha monic s uc u e
di ec ly om he in insic pe iodici y o he
g oup-delay ield, cha ac e ized by a single
uni e sal cons an 𝜏𝑔 and i s geome ic
p ojec ion. This pa ame e economy p o ides
causal pa simony: he obse ed esonance
pa e n a ises na u ally om he disc e e
p opaga ion o expansion delays wi hou
in oking non-ba yonic da k ma e , da k
ene gy, o an ini ial plasma epoch. The DCM
hus eplaces he mul i-componen acous ic
“ i ” o ΛCDM wi h a uni ied ha monic
in e p e a ion g ounded in he ela i is ic
delay mechanics o mass expansion.
The quan i a i e o mula ion o his ha monic
in e p e a ion is de eloped in Appendix C.10.
7. Conclusion
We ha e p esen ed he Disc e e Cosmology
Model (DCM), a amewo k ha
complemen s GR by a ibu ing cu a u e
and ime dila ion o ela i is ic g oup delays
in disc e ely expanding ma e . This
in e p e a ion upg ades he de ini ion o mass
wi hin he s ess–ene gy enso , p o iding a
causal–mechanical ounda ion a he han
ea ing mass as an unexplained sou ce e m.
DCM p ese es Eins ein’s equa ions while
en iching he sou ce sec o wi h delay e ms,
ensu ing conse a ion and consis ency wi h
es ablished geome y. The esul ing
amewo k yields h ee independen , es able
consequences: la galac ic o a ion cu es,
quad a ic supp ession o cosmological
edshi , and seismic–escape eloci y
con e gence. These p edic ions, especially
he seismic ela ion con i med by Apollo and
InSigh missions, dis inguish DCM om
phenomenological al e na i es such as
MOND o ΛCDM ex ensions.
By linking mic ophysical disc e eness
(Comp on-scale oscilla ions) o mac oscopic
as ophysical obse ables, DCM es ablishes
a b idge be ween ounda ional physics and
cosmology. This causal–mechanical
pe spec i e p o ides a no el, es able
app oach o he p oblems o da k ma e and
da k ene gy while p ese ing he s uc u e o
Gene al Rela i i y.
DCM uni ies o a ion cu es (RAR cu a u e
om a single 𝐻0-ancho ed scale), clus e
lensing ( ela i is ic 𝜎𝑣2 scaling), and
cosmological edshi supp ession.
The in e p e a ion ex ends na u ally o he
CMB, whose nea -pe ec iso opy a ises
om ho izon-shell e-emission a he han
om a p imo dial he mal epoch.
Al hough he p esen wo k ocuses on
g a i a ional and cosmological scales, whe e
cumula i e g oup delays domina e, he same
p inciple may mani es a he quan um scale
as disc e e delays, consis en wi h
hyd odynamic quan um analogs; a de ailed
19
ea men o spec al s uc u e, howe e , lies
beyond he scope o his pape .
Concluding Highligh s
• Seismic–g a i a ional law: A e age
seismic eloci ies con e ge wi h escape
eloci ies ac oss sel -g a i a ing bodies,
e ealing a new empi ical egula i y.
• Delay-based mechanism: G a i y and
cosmological edshi a ise om
cumula i e ela i is ic g oup delays in
disc e ely expanding ma e .
• Fla o a ion cu es: Galac ic dynamics
a e explained by combined g a i a ional
and kinema ic delays, wi hou in oking
da k ma e .
• Hubble ension: Quad a ic edshi
supp ession nea he cosmic ho izon
na u ally accoun s o he obse ed
disc epancy in H₀.
• Falsi iabili y: P edic ions can be es ed
wi h A emis luna seismology, galaxy
o a ion spec oscopy, and local-g oup
edshi su eys.
8 Fu u e Wo k
• Tes he seismic wa e co ela ion o
o he bodies wi h g a i y shaped
co es.
• Fu u e DCM es s may explo e s ella
bodies, p edic ing he Sun’s P-wa e
eloci y (~510 km/s) aligns wi h i s
escape eloci y (618 km/s, a io
~0.82) ia adial p ojec ion, es able
wi h ad anced helioseismology.
• Con i m Moon’s ci cula P-wa e
p opaga ion wi h A emis [14].
• Upscale Q-D i e a low empe a u es
[35].
Acknowledgemen s
The au ho hanks colleagues and
compu a ional ools o eedback and edi ing
suppo .
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APPENDIX A: GR-
Compa ible S ess-Ene gy
Tenso o he Disc e e
Cosmology Model (DCM)
The Disc e e Cosmology Model (DCM)
complemen s Gene al Rela i i y (GR) by
p o iding a causal-mechanical ounda ion
21
o he s ess-ene gy enso , in e p e ing
mass and cu a u e as eme gen om
disc e e in e ac ion delays. We main ain
Eins ein’s ield equa ions,
, (A1)
bu upg ade he sou ce e m o include
delay e ec s.
A.1 Two-Scale Link: Disc e e o
Con inuum
Le θ = ωCτ be he as Comp on phase, wi h
ωC he Comp on equency. The
mic oscopic enso τµνdisc(x,θ) encodes
phase-dependen mass ma(θ) and delay
s esses Dµν(x,θ) om ini e speed
in e ac ions (e.g., elec omagne ic s esses,
see Appendix A o he supplemen al
ma e ial). Unde scale sepa a ion ε =
(𝑡𝑠𝑦𝑠
−1 𝜔𝐶
⁄) ≪ 1 (e.g., ε ∼ 10-20 o plane a y
co es, ≪ 10-30 o galaxies), he
mac oscopic enso is:
,
(A2)
Ensu ing . Empi ically, he
seismic–escape eloci y con e gence
(Table 1, §2.3) calib a es he delay scala as
⟨Wco e⟩ ≃ e2/c2, linking disc e e dynamics
o mac oscopic cu a u e.
A.2 Exchange o m and o al
conse a ion
The s ess-ene gy enso is:
𝑇𝜇𝜈
DCM =𝑇𝜇𝜈
(ba )+Δ𝑇𝜇𝜈
(delay), (A3)
whe e he ba yon enso is:
𝑇𝜇𝜈
(ba )=(𝜌𝑏+𝑝𝑏/𝑐2)𝑢𝜇𝑢𝜈+𝑝𝑏𝑔𝜇𝜈 +
𝑞µ𝑢𝜈+𝜋𝜇𝜈, (A4)
And he delay enso is:
Δ𝑇𝜇𝜈
(delay)=𝜌𝑑𝑐2𝑢𝜇𝑢𝜈+𝑝𝑑ℎ𝜇𝜈 +𝜋𝜇𝜈
(𝑑), (A5)
wi h uµ he 4- eloci y, 𝑢𝜇𝑢𝜇=−1, ℎ𝜇𝜈 =
𝑔𝜇𝜈 +𝑢𝜇𝑢𝜈, 𝑞𝜇𝑢𝜇=0, 𝜋𝜇
𝜇=𝜋𝜇
(𝑑)𝜇=
0, (A6)
𝜋𝜇𝜈𝑢𝜈=𝜋𝜇𝜈
(𝑑)𝑢𝜈=0. (A7)
We allow exchange ia a 4- o ce densi y
𝑄𝜈(Fig.A.1):
∇𝜇𝑇𝜇𝜈
(ba )=−𝑄𝜈, ∇𝜇Δ𝑇𝜇𝜈
(delay)=+𝑄𝜈
⇒ ∇𝜇𝑇𝜇𝜈
DCM =0. (A8)
The weak- ield closu e used in disks is:
𝑊=𝜒𝑔Φba
𝑐2+𝜒𝑘𝑣2
𝑐2, (A9)
Ψkin(𝑟)=∫ 𝑣𝑐2(𝑠)
𝑠
𝑟
𝑟0 𝑑𝑠,
𝑄𝜈=𝜌𝑏 ∇𝜈(𝜒𝑘Ψkin). (A10)
The exchange ep esen s he ini e-speed
“delay s ess” needed o suppo o a ion;
𝜒𝑔,𝜒𝑘∼𝑂(1) and a e calib a ed
empi ically, no uni e sal cons an s.
A he mic oscopic le el, 𝑄ν ep esen s
momen um ans e om ini e-speed
22
Comp on-scale expansion (𝜀=𝜆𝐶/𝐿≪1);
he mac oscopic exchange law (A.6) is he
ensemble a e age o e hese disc e e delays.
Figu e A.1: Ba yons 𝑇𝜇𝜈
(ba )and delay sec o 𝑇𝜇𝜈
(delay)exchange ou -
o ce 𝑄ν. The exchanges cancel in he di e gence, ensu ing
∇𝜇𝑇𝜇𝜈
DCM =0 while allowing ini e-speed delay s esses o suppo
o a ion and edshi e ec s.
Thus, 𝜒𝑔 and 𝜒𝑘 do no in oduce new
uni e sal cons an s bu ins ead e lec
obse a ional unce ain ies (e.g. mass- o-
ligh a ios and ba yonic p o ile sca e ) when
coa se-g ained o e galac ic o plane a y
scales.
A.3 S a iona y, axisymme ic disks:
i e a i e closu e
P ojec ing ∇𝜇𝑇𝜇𝜈
(ba )=−𝑄ν adially o a
cold disk:
𝑣𝑐
2
𝑟=𝜕𝑟Φba +𝑄𝑟
𝜌𝑏=𝜕𝑟(Φba +𝜒𝑘Ψkin)≡
𝜕𝑟Φe . (A11)
To a oid ci cula i y, we sol e sel -
consis en ly:
1. Ini 𝑣𝑐(0): ba yons only, (𝑣𝑐(0))2/𝑟=
𝜕𝑟Φba .
2. Upda e Ψkin
(𝑛)(𝑟)=
∫(𝑣𝑐(𝑛−1)(𝑠))2
𝑟
𝑟0/𝑠 𝑑𝑠.
3. E ec i e Φe
(𝑛)=Φba +𝜒𝑘Ψkin
(𝑛) .
4. Veloci y (𝑣𝑐(𝑛))2/𝑟=𝜕𝑟Φe
(𝑛).
5. I e a e o |𝑣𝑐(𝑛)−𝑣𝑐(𝑛−1)|/|𝑣𝑐(𝑛−1)|<
𝛿 (e.g., 10−3).
As a oy example, o an exponen ial disk
wi h Σ𝑏(𝑟)=Σ0𝑒−𝑟/𝑅𝑑, wi h Σ0=108 𝑀⊙/
kpc2 and 𝑅𝑑=3 kpc, he i e a ion con e ges
a e ou s eps o 𝑣𝑐≈150 km/s a 𝑟≈
10 𝑘𝑝𝑐. This demons a es ha he sel -
consis en closu e ep oduces la o a ion
wi hou nulling he ba yonic po en ial Φba .
This con e gen closu e yields la ou e
segmen s wi hou nulling Φba ; o an
exponen ial disk i asymp o es o an
iso he mal-like ail.
A.4 Cosmology (FRW): iso opy and
con inui y
On FRW 𝑢𝜇=(1,0,0,0)0), shea - ee)
equi e π𝜇𝜈
(𝑑)=0:
Δ𝑇𝜇𝜈
(delay)=𝜌𝑑𝑐2𝑢𝜇𝑢𝜈+𝑝𝑑ℎ𝜇𝜈. (A12)
Wi h 𝐶≡−𝑢𝜈𝑄𝜈,
𝜌𝑏
+3𝐻(𝜌𝑏+𝑝𝑏/𝑐2)=−𝐶, (A13)
𝜌𝑑
+3𝐻(𝜌𝑑+𝑝𝑑/𝑐2)=+𝐶, (A14)
and ( o p ese e iso opy) ake 𝑄𝜈=𝐶 𝑢𝜈
(ene gy exchange only). The e a e wo
closu es:
23
• Conse a i e 𝐶=0, 𝑤𝑑≃−1+
𝑂(𝜀).
• Algeb aic 𝜌𝑑=3𝜀𝐻2/(8𝜋𝐺) wi h
𝜀≃0.08−0.10 (quad a ic edshi
supp ession used in §2.5).
The seismic law 𝑣𝑠2≃𝑣𝑒𝑠𝑐
2≃𝑐2𝑊 p o ides
an independen calib a ion o he delay
scala , ein o cing ha 𝜀≪1 b idges mic o-
le el disc e eness and mac o-scale
obse ables in bo h plane a y in e io s and
cosmological expansion.
A.5 Two-scale ke nel and iso he mal
ail
We de ine a minimal wo-scale ke nel ac ing
on 𝑣2:
𝐾(𝑟,𝑟′)=𝜒𝑔 𝛿(𝑟−𝑟′)
𝑟′+𝜒𝑘 Θ(𝑟−𝑟′)
𝑟𝑟′, (A15)
𝑔del(𝑟)=∫𝐾(𝑟,𝑟′) 𝑣2(𝑟′)
𝑟′ 𝑑𝑟′. (A16)
This p oduces Ψkin ∼ln𝑟 o e la segmen s
and
ρ𝑑(𝑟)=1
4𝜋𝐺𝑟2𝑑
𝑑𝑟[𝑟𝑣𝑐2]∝𝑟−2, (A17)
i.e. an iso he mal-like en elope wi hou da k
halos.
A.6 Compa ison o O he Theo ies
Unlike MOND, which in oduces an
empi ical accele a ion scale, DCM de i es
la o a ion cu es om kinema ic delays
wi hou ad hoc pa ame e s. Unlike scala -
enso heo ies (e.g., TeVeS), DCM’s delay
scala W is empi ically calib a ed by seismic
da a (Table 1), g ounding i in obse able
phenomena
As implemen ed in Appendix A.7: ba yonic
band om SMD-F/SMD-S; sel -consis en
i e a ion in 𝑣𝑐 and 𝑣𝑒; ensemble band c oss
{Υ∗,𝜒𝑔,𝜒𝑘}∼𝑂(1) e lec ing obse a ional
unce ain ies (no a MOND-like uni e sal
pa ame e ).
A.7 P edic ion Algo i hm o
Galac ic Ro a ion Cu es
The delay-based s ess–ene gy o mula ion
can be ope a ionalized in o a ep oducible
algo i hm o p edic ing galaxy o a ion
cu es om pho ome ic mass maps:
1. Ba yonic baseline: Su ace b igh ness
p o iles 𝑆𝑏(𝑅) a e con e ed o s ella
su ace densi ies using ca alog 𝑀/𝐿. Two
limi ing cases a e conside ed:
2. Ini ial eloci ies: An ini ial 𝑣𝑐(𝑅) is
o med by combining ba yonic
componen s.
3. Delay ke nel: The wo-scale delay
ope a o (Appendix A.3) is applied o 𝑣2,
yielding an e ec i e delay accele a ion
ield 𝑔del(𝑅).
4. I e a ion: 𝑣𝑐2=𝑅(𝑔ba +𝑔del) is
upda ed i e a i ely un il con e gence o
bo h c _c c and he associa ed escape
eloci y 𝒗𝒆𝒔𝒄.
5. Ensemble band: Pa ame e s (𝐿1/ℎ,𝐿2/
ℎ,χ1,χ2) a e scanned wi hin o de -uni y
anges. Models wi hin 10% o he bes
RMSE ela i e o obse ed 𝑣obs a e
e ained, de ining a p edic i e band.
24
This p ocedu e p oduces a amily o
o a ion cu es consis en wi h he
obse ed la ou e p o ile wi hou
in oking da k ma e halos. Figu e A.2
illus a es he me hod o galaxy U14,
showing he ba yonic band [11-12], he
DCM band, and he obse ed eloci ies
[13-14].
Figu e A.2: DCM p edic ion o o a ional eloci ies (UGC 14)
The p oposed algo i hm he e is no a i in he
MOND sense (no ee uni e sal 𝑎0) bu a
sel -consis en closu e o he delay enso
wi h empi ical ba yons.
A.8 Causal Simila i y be ween
Plane a y, Galac ic, and
Cosmological Accele a ion G adien s
The ee- all accele a ion p o ile inside a
sel -g a i a ing body e eals how
g a i a ional delay accumula es wi h adius.
In he Ea h's in e io , as shown in Fig. A.3,
he accele a ion 𝑔(𝑟) inc eases nea ly
linea ly h ough he co e, indica ing ha he
local g oup-delay ield builds up uni o mly
wi h dis ance om he cen e . Each shell
con ibu es cohe en ly o he cumula i e
dila ion g adien , p oducing a nea ly cons an
causal inc emen pe adial s ep. This egime
co esponds o a delay-sa u a ed domain in
which s ess p opaga ion and g a i a ional
dila ion ollow he same ela i is ic limi .
Beyond he co e, in he man le and c us ,
densi y and igidi y a ia ions in oduce
discon inui ies, and 𝑔(𝑟) becomes
i egula —signi ying in e e ence be ween
pa ially decoupled delay pa hways.
Figu e A.3: Ea h's g a i y acco ding o he P elimina y Re e ence
Ea h Model (PREM) [16]
A simila causal opology appea s in galac ic
sys ems (Fig. A.2). Wi hin he galac ic co e,
he obse ed o a ion eloci y ises
app oxima ely linea ly wi h adius (𝑣 ~𝑟),
implying a linea accele a ion p o ile
𝑔(𝑟)~𝑟, analogous o he plane a y-co e
egime. He e, he g oup-delay ield
accumula es cohe en ly ac oss s ella shells,
main aining a uni o m delay g adien and
s able causal coupling. In con as , a la ge
adii, whe e he disk ansi ions o he halo,
g a i a ional accele a ion la ens o
oscilla es. The co esponding delay ield
becomes agmen ed by o a ional shielding
and oid asymme ies, p oducing quasi-
s a iona y in e e ence be ween disc e e
expansion shells. This ansi ion om
cohe en o in e e en ial delay beha io
explains bo h he la ening o galac ic
25
o a ion cu es and hei sensi i i y o
mo phology, wi hou in oking da k ma e .
Thus, he DCM in e p e s plane a y and
galac ic accele a ion s uc u es as
mani es a ions o he same unde lying
p inciple: a cohe en linea buildup o
ela i is ic g oup delay in he cen al egions,
ollowed by e a ic o esonan delay
in e e ence in he ou e zones. This causal
sel -simila i y ac oss scales suppo s he
uni e sali y o delay mechanics in shaping
bo h g a i a ional and kinema ic phenomena.
The same delay-g adien pa e n ex ends o
he la ges scale o s uc u e. The linea
Hubble ela ion 𝑣=𝐻0𝑟 ep esen s he
cosmological analogue o he co e- egime
cohe ence, whe e g oup delays accumula e
uni o mly ac oss space. This egime de ines
he global causal expansion ield o DCM—
he cosmic equi alen o he uni o m
accele a ion zone in plane a y and galac ic
in e io s. A g ea e sepa a ions, nea he
obse able ho izon, his cohe ence becomes
agmen ed by disc e e delay shells, gi ing
ise o quasi-ha monic esonances obse ed
in he cosmic mic owa e backg ound. Thus,
om plane a y co es o he Hubble ho izon,
all sel -g a i a ing sys ems exhibi he same
sequence: cohe en linea buildup o delay
ollowed by disc e e in e e ence, go e ned
by one uni e sal delay-mechanics p inciple.
A he la ges scale, he linea i y o he
Hubble ela ion 𝑣=𝐻0𝑟 ep esen s he
cosmological mani es a ion o his same
delay-g adien cohe ence, comple ing he
causal hie a chy om plane a y co es o he
expanding Uni e se.
APPENDIX B:
Obse e ‑Local Fac o s
and Ho izon Relay Lemma
B.1 No-local-cap lemma
Le (1+𝑧𝑜𝑏𝑠)=𝐶𝑙𝑜𝑐 ·(1+𝑧𝑝𝑎𝑡ℎ) wi h
cons an 𝐶𝑙𝑜𝑐 >0. I 𝑙𝑖𝑚𝑟→𝑅𝐸𝐻(1+
𝑧𝑝𝑎𝑡ℎ)=∞ (ha d ho izon), hen
𝑙𝑖𝑚𝑟→𝑅𝐸𝐻(1+𝑧𝑜𝑏𝑠)=∞ and 𝑑𝑧𝑜𝑏𝑠/𝑑𝑟=
𝐶𝑙𝑜𝑐 ·𝑑𝑧𝑝𝑎𝑡ℎ/𝑑𝑟. Thus a cons an local ac o
canno p oduce a ini e 𝑧𝑚𝑎𝑥 no en o ce
𝑑𝑧/𝑑𝑟→0.
B.2 Relay (pene a ion) cla i ica ion
Pho ons o igina ing beyond Ea h’s ho izon
do no a i e a Ea h in ini e obse e ime.
An obse e who eloca es ou wa d can
ecei e hose pho ons a hei new loca ion
because hei pe sonal ho izon mo es;
in o ma ion can hen be elayed back o Ea h
ia new local emission, bu he o iginal
wa e on s ha e no c ossed Ea h’s ho izon.
Hence obse e ‑ ela i e ho izons a e
consis en : isibili y di e s by loca ion
wi hou con adic ion.
APPENDIX C: Cosmic
Backg ound and Ho izon-
Shell Pola iza ion
C.1. O igin o he Backg ound Field
In he Disc e e Cosmology Model (DCM),
he cosmic mic owa e backg ound (CMB) is
no in e p e ed as elic adia ion om a
p imo dial ho epoch, bu as a s a iona y e-
emission phenomenon a ising a he isibili y
32
Tes abili y: look o phase‑synch onized
bi h e en s nea nuclei; sea ch o ansien ,
s ep‑wise g ow h signa u es in ul a as
pump–p obe expe imen s; es p o on-
posi on beams o a ac ion, il e ing
magne ic e ec s; measu e unneling a es in
STM/quan um wells unde a ying ields
(p edic s mass luc ua ions) [31]. These a e
high‑ isk, high‑ ewa d es s in en ionally
seg ega ed om he g a i a ional/
cosmological co e o he manusc ip o a oid
o e each.
APPENDIX F:
Gy omagne ic Ra io om
Phase-Weigh ed Ine ia
(Toy Model)
Assume cha ge ci cula es du ing a ac ion
o a Comp on cycle wi h o a ing mass m o
and adius . Fo a ing (κ =1),
𝜇≈𝑞
2𝑓𝑟2𝜔,𝑆≈𝑓𝑚𝑟𝑜𝑡𝑟2𝜔
⇒ 𝜇
𝑆≈𝑞
2𝑚𝑟𝑜𝑡 (F1)
Iden i ying 𝜇≈𝑔 𝑞
2𝑚0S yields 𝑔 ≈
𝑚0/𝑚𝑟𝑜𝑡. I 𝑚𝑟𝑜𝑡 = 𝑚0/2, hen 𝑔 ≈2
wi hou supe luminal o a ion. Small phase
asymme ies and EM sel -in e ac ions yield a
na u al 𝑔−2 co ec ion dependen on he
delay scala 𝑊.
APPENDIX G - S a is ical
Consis ency Ac oss Scales
Table 4: Summa y o 𝛬𝐷𝐶𝑀 =𝑎𝑜𝑏𝑠/(𝐻0
2𝑅/
2) ac oss sys ems.
Sys em
Scale (m)
𝚲𝐃𝐂𝐌
Unce ain y
(±)
Plane a y
(Ea h, Ma s,
Venus)
106–107
1.00
0.10
Luna /
Tidally
locked
106
1.05
0.12
Galac ic
(SPARC)
1019–
1021
0.98
0.08
Cosmological
(SN Ia, BAO)
1024–
1026
1.00
0.05
As e oidal
(undi .)
103–104
>1.5
0.30
APPENDIX H: Resonan
O e shoo Hypo hesis o
Lep on Masses
In he Disc e e Cosmology Model (DCM),
pa icle mass luc ua es disc e ely a
Comp on equencies. While he elec on
ep esen s he s able baseline o his cycle,
hea ie lep ons may be unde s ood as
esonan o e shoo s a es ha occu when
disc e e delays accumula e cohe en ly.
H.1 A Toy De i a ion o he
Muon/Elec on Mass Ra io
In he Disc e e Cosmology Model (DCM),
mass oscilla es a Comp on equency. The
elec on co esponds o he s able baseline,
while he muon a ises as he i s cohe en
esonan o e shoo s a e.
Two delay channels con ibu e pe Comp on
cycle:
33
• a longi udinal (g a i a ional-like)
channel, linea in delay quan a,
• a ans e se (kinema ic) channel,
weigh ed by ½β² as in he weak- ield
expansion o γ.
DCM p edic s ha he i s o e shoo occu s
when hese channels close cohe en ly. The
e ec i e muon/elec on a io ollows om
combining hem:
(𝑚𝜇/𝑚𝑒)≈𝛼−1 +½𝛼−1 =1.5𝛼−1 (H1)
Nume ically, wi h 𝛼−1 =137.036:
(3/2)𝛼−1 =205.554 (H2)
The obse ed alue is:
𝑚𝜇/𝑚𝑒=206.768 (H3)
lea ing a small esidual co ec ion
𝛿=(𝑚𝜇/𝑚𝑒)−(3/2)𝛼−1 ≈1.214 (H4)
Thus, he leading e m explains 99.4% o he
a io wi hou ee pa ame e s, while he
esidual is plausibly due o mic o-le el
ine ia e ec s.
H.2 Rela ing he Co ec ion δ o
Phase-Weigh ed Ine ia
Appendix F p oposed a phase-weigh ed
ine ia oy model: du ing each Comp on
cycle, a ac ion o he mass o a es (m o )
and he es expands (mexp). The cycle-
a e aged ine ia is
𝑚 =𝑓𝑚𝑟𝑜𝑡+(1−𝑓)𝑚𝑒𝑥𝑝 (H5)
Fo he elec on baseline, he o a ing mass
sa is ies 𝑚𝑟𝑜𝑡
(𝑒)≈𝑚0/𝑔𝑒, wi h 𝑔𝑒≃2(1+
𝛼/2𝜋). De ine he o a ional sha e:
𝜒=𝑚𝑟𝑜𝑡
(𝑒)/𝑚(𝑒) (H6)
A he i s esonance, le he o a ing mass be
ampli ied by a ac o η:
𝑚𝑟𝑜𝑡(𝜇)=𝜂𝑚𝑟𝑜𝑡(𝑒) (H7)
The muon/elec on ine ia a io is hen:
𝑚(𝜇)/𝑚(𝑒)=1+(𝑓𝜒/𝑔𝑒)(𝜂−1) (H8)
Iden i ying his small ac o wi h he addi i e
co ec ion δ:
𝛿≈(𝑓𝜒/𝑔𝑒)(𝜂−1) (H9)
Wi h ypical alues 𝑓 ≈ 0.4 − 0.6, 𝜒 ≈
0.3 − 0.6, and 𝑔𝑒≈2.0023, he obse ed
δ ≈ 1.21 is ep oduced i he esonance
ampli ies he o a ing-phase ine ia by a
modes ac o 𝜂 ≈ 7 − 15. This o de - en
enhancemen is physically easonable wi hin
he DCM pic u e, whe e he o e shoo
comp esses he e ec i e o a ional a c o he
Comp on cycle.
The small co ec ion 𝛿≈1.21 a ises in he
phase-weigh ed ine ia model (Appendix F).
In his pic u e, he o a ing ac ion o he
elec on’s mass is ampli ied a esonance by
a ac o η≈7−15, consis en wi h modes
Comp on-scale dis o ions. This p o ides a
na u al basis o he 𝛿 e m. Expe imen ally,
such an ampli ica ion could lea e signa u es
in p ecision muon 𝑔−2 measu emen s,
o e ing a po en ial es o he DCM
amewo k.
Summa y:
• Leading e m: (3/2) α⁻¹ = 205.554
(pa ame e - ee).
• Co ec ion: δ ≈ 1.21, explained by
phase-weigh ed ine ia.
34
• Resul : Combined, hese gi e 𝑚𝜇/𝑚𝑒≈
206.8, in ag eemen wi h expe imen .
This sugges s ha lep on mass a ios eme ge
om disc e e expansion dynamics and
esonan closu e condi ions, a he han being
independen inpu s o he S anda d Model.
H.3 Second esonance ( au) as a
cons ained, no ye de i ed, s a e
S a emen o ac s.
The expe imen al alue is 𝑚𝜏/𝑚𝑒≈
3477.23. Ou muon logic ( wo-channel
closu e yielding α−1+1/2𝛼−1 does no
i ially gene alize: a nai e “5/2𝛼−1”
leading e m unde shoo s by an o de o
magni ude and is he e o e ejec ed. We ea
au as a second cohe en esonance whose
mic o-closu e di e s om he muon’s.
Minimal cons ain s DCM imposes (no
i ing):
1. Di e en closu e opology. The
second esonance mus use a dis inc
channel composi ion om muon
(e.g., mul iplica i e/compound
closu e o addi ional sel -ene gy
channel), o he wise he scale s ays
𝒪(α−1), no 𝒪(103).
2. Single-scale o igin. No new
uni e sal cons an s beyond α alphaα
and he same DCM mic o-physics
(phase-weigh ed ine ia) a e allowed;
la ge ac o s mus eme ge om
esonance o de (compound
phasing), no om ad hoc pa ame e s.
3. Li e ime o de . Highe esonance ⇒
na owe window ⇒ much sho e
li e ime (consis en wi h 𝜏’𝑠 s scale
s 𝜇’𝑠 μs).
4. Con inui y wi h muon co ec ion.
The small addi i e co ec ion
mechanism δ∼𝑓χ
𝑔𝑒(η−1) should
scale p edic ably wi h esonance
o de (e.g., wi h an o de pa ame e
𝑁)—no be e- uned. (H.2)
Two conc e e candida e s uc u es o
explo e (do no claim sol ed):
• Compound closu e (p oduc o m).
Ins ead o adding channels, he second
esonance could equi e sequen ial closu es
in he same cycle, gi ing an e ec i e
ampli ica ion p opo ional o a p oduc o
i s -o de ac o s. Schema ically,
𝑚(2)
𝑚𝑒 ∼ (3/2α−1) 𝒬(𝑁,α) (H10)
whe e 𝒬 is a esonance-o de mul iplie
om compound phasing (e.g., du y-cycle
compac ion and sel -ene gy eweigh ing
ac oss wo closu es). This na u ally
p oduces 𝒪(101-2) mul iplie s wi hou new
cons an s. (Quan i a i e de i a ion TBD.)
As a schema ic placeholde , we w i e
𝑚τ
𝑚𝑒 ∼ (3/2𝛼−1) ℱ(𝑁=2), (H11)
whe e ℱ(2) ep esen s he compound-
closu e o adia i e channel ac o . The
empi ical alue 𝑚𝜏/𝑚𝑒≈3477 sugges s
ha ℱ(2) is an o de - en mul iplie ela i e o
he muon case. A mic ophysical de i a ion o
ℱ(𝑁,𝛼) is le as a p og am o u u e wo k.
• Thi d channel pa icipa ion. The second
esonance migh ac i a e an
elec omagne ic sel -in e ac ion channel
( adia i e e m) cohe en ly wi h he
longi udinal/ ans e se pai . I s inclusion a
35
esonance o de 𝑁=2 could boos he
scale o 𝒪(103) while keeping he muon’s
𝛿 mechanism in ac (H.2), i.e.,
𝑚𝜏/𝑚𝑒 ≈ (3/2∝−1) 𝑅(𝑁= 2,𝛼) + 𝛿𝜏, (H12)
𝛿𝜏 om he same (𝑓,χ,𝑔𝑒,η) law wi h N-
scaling.
Wha we don’ do:
We do no p esen a numbe o 𝑚𝜏/𝑚𝑒 om
a simplis ic channel sum. Ins ead, we ele a e
au o a a ge o a o hcoming mic o-
de i a ion ha uses he same machine y as
H.2 (phase-weigh ed ine ia) bu ex ended o
compound closu es and/o a hi d channel.
Falsi iable o ecas (band, no a poin ):
Once a speci ic compound-closu e ule is
chosen, i mus :
• educe o H.1 o 𝑁=1 (muon),
• keep he same 𝛿-law modulo an
explici 𝑁-scaling, and
• hi 𝑚𝜏/𝑚𝑒 wi hin a na ow,
pa ame e - ee leading band, wi h
δτ del a_ auδτ ixed by he same
mic o-co ec ion s uc u e (no e i ).
(Edi o ial no e o e iewe s: his sec ion
explici ly acknowledges he cu en
limi a ion and se s a es able p og am a he
han e o i ing numbe s.)
We in e p e he au as a second-o de
esonance equi ing compound closu e; we
ou line cons ain s and a alsi iable p og am
bu de e a nume ic de i a ion o u u e wo k.
H.4 — Shell-Laye ed Resonance and
Lep on S abili y
In DCM, he elec on’s e ec i e en elope is
se by one educed Comp on wa eleng h,
𝜆𝐶
=ℏ
𝑚𝑒𝑐, (H13)
since each Comp on cycle can expand he
in e ac ion ield by a mos 𝑐Δ𝑡=𝜆𝐶
. We
in e p e his en elope as he cumula i e
ex en o he pa icle’s delay ield.
Shell-laye ing mechanism
Highe esonances (muon, au) a ise as
cohe en o e shoo s in which addi ional
Comp on-scale shells a e s acked. Each shell
co esponds o he ac i a ion o an addi ional
delay channel:
• Elec on (𝑵=𝟎): baseline shell.
• Muon (𝑵=𝟏): longi udinal +
ans e se channels close cohe en ly,
yielding one ex a shell.
• Tau (N = 2): equi es compounded
closu e wi h a adia i e sel -
in e ac ion channel, s acking ye
ano he shell.
Thus, he e ec i e en elope g ows as
𝑅𝑁≈(𝑁+1) 𝜆𝐶
, (H14)
wi h each shell indi idually cons ained by
he ligh -speed limi . The obse ed lep on
gene a ions co espond o he i s h ee such
closu es.
36
Mass a ios and shells
The muon a io ollows om he wo-channel
closu e (H.1):
𝑚μ
𝑚𝑒 ≈ 3/2 α−1+δ, (H15)
δ≈1.21.
The au can hen be in e p e ed as he second-
o de closu e (𝑁=2) equi ing
compounded shells. A gene al ansa z
consis en wi h H.1–H.3 is:
𝑚𝑁
𝑚𝑒 ≈ 3/2 α−1 𝒬(𝑁,α), (H16)
wi h 𝒬(𝑁=1)=1 (muon) and 𝒬(𝑁=
2)∼2α−1/3 ( au), gi ing he obse ed ∼
3477. De i ing 𝒬(𝑁,α) om mic ophysics
emains a ask o u u e wo k, bu he shell
amewo k p o ides a na u al sca old.
Li e imes om cohe ence decay
Each addi ional shell inc eases phase
complexi y and educes s abili y. We model
he cohe ence li e ime as
τ𝑁 ∼ τ0
(𝑁+1)2, τ0≈10−21 s, (H17)
he Comp on imescale. This scaling yields
(Fig. H.1):
• 𝑁=0 (elec on): τ0→ ∞.
• 𝑁=1 (muon): τ1∼2 μs.
• 𝑁=2 ( au): τ2∼0.3 ps.
These alues a e consis en wi h obse ed
lep on li e imes (𝜇: 2.2 𝜇𝑠; 𝜏: 0.29 𝑝𝑠),
suppo ing he dual c i e ion o phase
closu e + cohe ence h eshold.
Figu e H.1: Lep on shells
Tes able p edic ions
• Fini e spec um: No highe lep ons
exis beyond au, as addi ional shells
collapse be o e o ming physical
s a es.
• En elope e ec s: P ecision
sca e ing nea Comp on scales could
e eal laye ed s uc u es in e ec i e
cha ge dis ibu ions.
• Anomalous g- ac o s: Muon and au
𝑔−2 should show de ia ions
consis en wi h al e ed delay- ield
en elopes.
• Scaling consis ency: The same shell-
laye ing logic unde pins bo h
mic oscopic lep on s uc u e and
mac oscopic plasma ou lows
(Appendix I).
Expe imen al es s could include sca e ing a
ene gy scales nea he elec on Comp on
wa eleng h ( ens o MeV), whe e mul i-shell
s uc u es migh lea e measu able de ia ions
in e ec i e cha ge dis ibu ions.
Al e na i ely, high-p ecision muon and au
𝑔−2 expe imen s could e eal anomalous
con ibu ions om laye ed delay en elopes.
H.5 Comp on-Scale Tes s and Lep on g-2
In he Disc e e Cosmology Model (DCM),
lep ons (elec on, muon, au) a e desc ibed as
37
esonan delay s a es a ising om cohe en
o e shoo s o disc e e expansion and o a ion
a he Comp on equency. Each lep on o de
adds a quan ized delay shell, al e ing he
phase s uc u e o he in e nal in e ac ion
ield and, consequen ly, i s magne ic momen
and sca e ing beha io .
Lep on Shell Hypo hesis.
• Elec on: Single baseline shell,
de ined by he educed Comp on
wa eleng h 𝜆𝐶,𝑒 ≃2.43×10−12 m.
• Muon: Second shell p oduced by
longi udinal– ans e se delay
closu e, inc easing ine ia by ∼206
(𝑚𝜇/𝑚𝑒=206.768).
• Tau: Thi d shell o med h ough
compounded adia i e closu e, 𝑚𝜏/
𝑚𝑒≈3477.
These laye s modi y he lep on’s e ec i e
cu en dis ibu ion and magne ic momen
h ough delay- ield en elopes, p oducing
small, quan ized de ia ions om he Di ac
limi 𝑔=2.
Connec ion o Muon 𝑔-2.
The anomalous magne ic momen 𝑎𝜇=(𝑔−
2)/2 p o ides a p ecision es o DCM’s
delay s uc u e. The S anda d Model (SM)
p edic s 𝑎𝜇
SM ≈116591810(43)×10−11,
while ecen measu emen s yield 𝑎𝜇
exp ≈
116592061(41)×10−11, a 4.2𝜎
disc epancy. In DCM, he second delay shell
adds a phase-weigh ed con ibu ion
𝑎ℓ
DCM ≃𝑎ℓ
SM+𝑁 𝛿(𝜏𝑔,𝛼), (H18)
whe e 𝛿 is a dimensionless mic o-co ec ion
(10−9–10−8), pending mic ophysical
de i a ion (see Appendix H.1), depending on
he g oup-delay cons an 𝜏𝑔 and he ine-
s uc u e cons an 𝛼. Fo he muon (𝑁=2),
his addi ional e m could shi 𝑎𝜇 by 10−8–
10−7, wi hin he p ecision ange o
Fe milab’s E989 expe imen . A consis en
excess ma ching his scale would suppo
DCM’s laye ed-delay in e p e a ion; absence
o such de ia ion would alsi y he lep on-
shell hypo hesis — and wi h-i DCM’s
Comp on-scale ex ension.
Comp on-Scale Sca e ing.
I lep ons possess disc e e shells, hei cha ge
o m ac o 𝐹(𝑞2) should exhibi weak
oscilla o y modula ions a ound 𝑞2∼1/𝜆𝐶,𝜇
2.
P ecision 𝑒–𝜇 sca e ing (a acili ies such as
Je e son Lab o u u e ILC expe imen s)
could p obe his egime:
𝑞𝜇
2≈1
𝜆𝐶,𝜇
2≈3.5×
1018 m−2 (equi alen o ∼
1 GeV2, p obing 100–200 MeV). (H19)
DCM p edic s 1–5% de ia ions in
di e en ial c oss-sec ions ela i e o he
SM’s smoo h all-o . De ec ion o such
oscilla o y s uc u e would e eal he shell
laye ing di ec ly; null esul s would exclude
i .
38
Expe imen al Implica ions.
Table 5: P edic ed obse ables o Comp on-
scale es s o he DCM lep on-shell
hypo hesis.
Expe imen
Obse able
DCM
Expec a ion
S a us /
Tes abili y
Muon 𝑔-2
(Fe milab
E989)
𝑎𝜇
=(𝑔
−2)/2
Δ𝑎𝜇=
10−8–
10−7
2025–26
un, ±0.1
ppm
p ecision
𝑒–𝜇
sca e ing
(JLab / ILC)
𝑑𝜎/𝑑Ω
s. 𝑞2
1–5%
oscilla o y
de ia ion
nea 1/
𝜆𝐶,𝜇
2
0.1%
p ecision
easible
(JLab 12
GeV)
𝜏 s udies
(BESIII /
Supe KEKB)
Fo m-
ac o
scaling
(𝑁=3)
La ge
delay-
phase shi
Fu u e,
explo a o y
In e p e a ion and Falsi iabili y.
The DCM amewo k hus p edic s
obse able signa u es o disc e e delay shells
a Comp on scales. Ag eemen be ween
measu ed 𝑎𝜇 excess o sca e ing oscilla ions
and he p edic ed delay magni ude would
subs an ia e he model’s mic o-causal
s uc u e. I , howe e , u u e da a con i m
S anda d Model alues wi hin expe imen al
unce ain y, he lep on-shell hypo hesis—and
wi h i , DCM’s Comp on-scale ex ension—
would be alsi ied. These es s p o ide a
di ec empi ical ou e o e i ying he delay-
shell mechanism in oduced in
Appendix H.1.
APPENDIX I —
Collisionless Plasmas,
S ella Winds, and
Rela i is ic Je s in DCM
In he Disc e e Cosmology Model, he
g a i a ional ield is a mac oscopic,
collec i e phenomenon a ising om
ela i is ic g oup delays wi hin cohe en ly
s uc u ed, ex ended mass domains (§2.2,
Appendix A). Single elemen a y pa icles and
uly collisionless plasmas possess no
in e nal delay s uc u e and he e o e
gene a e no g a i a ional delay ield o hei
own. Howe e , e e y es pa icle —
collisional o collisionless, massi e o
massless — mo es on geodesics o he
e ec i e me ic gene a ed by he cumula i e
delay ield o he mac oscopic cen al body
(Eq. 6). The equi alence p inciple and
ene gy–momen um conse a ion a e
p ese ed a he e ec i e le el (Appendix A).
The sub le asymme y (s uc u ed bodies
sou ce he ield; uns uc u ed pa icles only
ollow i ) has pa icula ly sha p
consequences o sys ems in which he
ou lowing ma e ial apidly becomes
collisionless: - Ho -s a winds and co onal
mass ejec ions - Plane a y pola /au o al
winds - Rela i is ic je s om AGN, X- ay
bina ies, and gamma- ay bu s s In hese
en i onmen s he ou wa d impulse ( adia ion
p essu e, wa e/ u bulence hea ing, magne ic
o ques, Bland o d–Znajek ex ac ion, e c.)
encoun e s almos ze o in e nal iscosi y and
ze o abili y o he ou lowing plasma o
gene a e a coun e -delay (i.e., g a i a ional)
ield. The esul is nea -pe ec con e sion o
39
deposi ed ene gy in o di ec ed bulk kine ic
ene gy — na u ally explaining:
1. Te minal wind eloci ies ou inely
eaching 𝑣∞≈ 2 − 5 𝑣𝑒𝑠𝑐 in O/B/WR
s a s (obse ed 1.5–5; [36, 37]).
2. The long-s anding “weak-wind p oblem”
and he di icul y o line-d i en wind
models o o e -p edic e minal speeds
wi hou ad-hoc clumping o po osi y
co ec ions.
3. The ex eme e iciency and collima ion
o ela i is ic je s (Lo en z ac o s Γ ≳
10–100, adia i e e iciencies >50–100%
o acc e ed es mass in some blaza s and
GRBs) [38, 39].
These ea u es a e quali a i ely and
quan i a i ely mo e na u al in DCM han in
s anda d GR+MHD, whe e collisionless
pa icles emain ully bound by he deep
g a i a ional po en ial un il su icien non-
g a i a ional o ces a e supplied. In DCM he
e ec i e sel -binding o he ou lowing
plasma is absen , since collisionless pa icles
canno sus ain an in e nal delay ield; only
he cen al body con ibu es o he me ic.—
a clea , alsi iable signa u e. High- esolu ion
UV/X- ay spec oscopy (XRISM, A hena,
Lynx) and u u e in-si u p obes o s ella -
wind accele a ion egions o je -launching
zones will di ec ly es which desc ip ion is
co ec [40, 41]. Thus, he mos powe ul
s ella winds, plane a y pola ou lows, and
ela i is ic je s cons i u e sensi i e na u al
labo a o ies o he eme gen , g oup-delay
o igin o g a i y p oposed by DCM.
