Vib a ional Wa e Dispe sion Hypo hesis
(HDOV):
Uni ied F amewo k, Mas e Equa ion, and
Mul idomain Applica ions
A noldo Wal e Fe nández
[email p o ec ed]
No embe 25, 2025
P ep in — Zenodo
Abs ac
We p esen a uni ied manusc ip o he HDOV o malism (Vib a ional Wa e Dis-
pe sion Hypo hesis), which p oposes an e ec i e mas e equa ion o wa e p opaga-
ion in en i onmen s wi h limi ed unc ional accessibili y. S a ing om a co a ian
ac ion wi h an en i onmen al ga e ac o (1 + 2g χ(I)ηp) ha modula es he kine ic
e m o a scala ield Ψ, we de i e a p ojec i e wa e equa ion which, in he high-
equency and la -space limi , educes o a modi ied Helmhol z equa ion. We show
how he ib a ional accessibili y ηpand he ga e χ(I)ac as an e ec i e complex
e ac i e index whose eal pa shi s he dispe sion ela ion and whose imagina y
pa implemen s unc ional a enua ion wi hou in o ma ion des uc ion.
In he eikonal egime, we de i e a WKB-like anspo law o he ampli ude
A(λ), which ac o izes in o a pu ely ela i is ic ocusing e m and an addi ional ex-
ponen ial ac o associa ed wi h accessibili y. On his basis, we build an ope a ional
dic iona y ha allows he same o mal scheme o be applied o seemingly dispa a e
domains: ime- esol ed X- ay sca e ing (TRXS) in molecula sys ems, egime an-
si ions a he heliopause (Voyage 1 da a), a ia ions in he leng h o day (LOD)
and e ec i e cosmic accele a ion (Pan heon+ and BAO).
In each domain we summa ize quan i a i e i s, o de s o magni ude and ele an
unc ional pa ame e s, and discuss alsi iable p edic ions associa ed wi h he s uc-
u e o unc ional accessibili y and i s impac on speci ic obse ables. The aim o
his manusc ip is o consolida e, in a single cohe en amewo k, esul s p e iously
p esen ed in sepa a e wo ks and o es ablish a clea basis o u u e obse a ional
and expe imen al es s.
1
Con en s
1 In oduc ion 3
2 The Uni ied HDOV F amewo k: Ac ion P inciple and Mas e Equa ion 3
3 De i a ion om he Modi ied Helmhol z Equa ion 4
3.1 S anda d Helmhol z equa ion and i s e ec i e modi ica ion ......... 4
3.2 Physical in e p e a ion o he unc ional e m ................ 4
3.3 Reco e y om he co a ian mas e equa ion ................. 5
4 WKB Regime and T anspo Law o he Ampli ude 5
5 Ope a ional Dic iona y by Domain 6
6 Applica ions and Mul idomain Valida ions 6
6.1 Quan um Domain: Func ional Modula ion in TRXS ............. 6
6.2 Heliosphe ic Domain: The Heliopause and Voyage ............. 8
6.3 Geodynamics Domain: Co e–Man le Coupling ................ 11
6.4 Cosmological Domain: Expansion and Ringdown ............... 12
6.5 Summa y o Mul idomain Func ional Valida ions .............. 14
7 Discussion: Eme gen Accessibili y and Hie a chical In a iance 16
No e on he S a us o he HDOV F amewo k 18
8 Conclusions and Falsi iable P edic ions 18
Appendix A — Func ional De i a ion o he T anspo Law 20
Appendix B — No a ion and Re e ence Uni s 22
2
1 In oduc ion
The HDOV hypo hesis p oposes ha wa e p opaga ion in egimes whe e he medium ex-
hibi s unc ional cons ain s — o example due o cu a u e, diso de , s ong g adien s o
hie a chical s uc u e— canno be desc ibed solely by linea wa e equa ions wi h cons an
coe icien s (Fe nandez 2025c; Fe nandez 2025e). Ins ead, i in oduces a ib a ional ac-
cessibili y me ic ηpand an en i onmen al ga e χ(I) ha modula e he e ec i e dynamics
o he ield.
The objec i e o his manusc ip is o uni y, wi hin a single ma hema ical ame-
wo k, se e al esul s p e iously p esen ed in sepa a e wo ks Fe nandez (2025e), Fe nan-
dez (2025d), Fe nandez (2025g), Fe nandez (2025 ), Fe nandez (2025b), and Fe nandez
(2025a), and o show how he same o mal s uc u e can:
•be de i ed om a co a ian ac ion p inciple;
• educe o a modi ied Helmhol z equa ion in he la -space limi ;
•gene a e a WKB anspo law wi h well-de ined unc ional a enua ion;
•be applied cohe en ly o domains as di e se as TRXS, he heliopause, geodynamics
and cosmology.
To his end, we o ganize he ex as ollows. In Sec ion 2we p esen he uni ied HDOV
amewo k s a ing om an e ec i e ac ion and de i e he p ojec i e mas e equa ion. In
Sec ion 3we show how, in he high- equency and la -space limi , one eco e s a modi ied
Helmhol z equa ion wi h an e ec i e complex e ac i e index. Sec ion 4discusses he
eikonal egime and he anspo law o he ampli ude. In Sec ion 5we build an ope -
a ional dic iona y ha connec s unc ional pa ame e s ac oss domains. Sec ions 6.1–6.4
p esen conc e e applica ions. Finally, in Sec ion 8we discuss implica ions and alsi iable
p edic ions.
Ve sion no e
In his i s e sion o he manusc ip , some o he igu es a e eused om he o iginal
Spanish HDOV pape s and s ill con ain axis labels and legends in Spanish. The quan i-
a i e con en , da ase s and i s a e exac ly he same as hose desc ibed in he ex , and
all cap ions a e p o ided in English.
A u u e e ised e sion ( 2) will include ully localised igu es wi h English labels,
wi hou changing he unde lying da a analysis.
2 The Uni ied HDOV F amewo k: Ac ion P inciple and
Mas e Equa ion
The s a ing poin is an e ec i e ac ion ha couples he dynamical ield Ψ o geome y
and o a ib a ional accessibili y me ic ηp, modula ed by an en i onmen al ga e χ(I). In
uni s whe e c=ℏ= 1, he e ec i e ac ion eads
S=Zd4x√−gM2
Pl
2R+Lma +1
21+2g χ(I)ηp∇µΨ∇µΨ−1
2m2Ψ2,(1)
whe e gis a dimensionless coupling, χ(I)is a ga e unc ion depending on in a ian s I( o
example, scala cu a u e, ield g adien s, mode densi y), and ηpis a posi i e scala ha
measu es he unc ional accessibili y o he medium.
3
Va ying he ac ion wi h espec o Ψyields he p ojec i e mas e equa ion:
∇µ(1 + 2g χ(I)ηp)∇µΨ+m2Ψ=0.(2)
This is a wa e- ype equa ion wi h a p ojec i e dispe sion e m con olled by χ(I)ηp. Fo
g→0o in en i onmen s whe e χ(I)ηp→0, one eco e s he s anda d Klein–Go don
equa ion. When χ(I)ηpbecomes signi ican , he e ec i e dynamics o Ψ e lec s a educed
unc ional accessibili y: some modes emain in he ope a ional subspace o he obse e ,
while o he s a e p ojec ed ou o i .
In his uni ied o mula ion, bo h ηpand χ(I)a e ea ed as e ec i e backg ound ields:
hey a e assumed o be eal unc ionals o he in a ian s o he medium and a e econ-
s uc ed om obse a ional da a in each domain. Tha is, in his wo k we do no ye
pos ula e an independen equa ion o mo ion o ηpand χ(I)de i ed om a sepa a e
a ia ional p inciple. A na u al dynamical gene aliza ion would consis in adding kine ic
and sel -in e ac ion e ms o ηp o he ac ion, he eby ob aining a coupled e olu ion equa-
ion o (Ψ, ηp)by join a ia ion, in he spi i o he ex ensions ske ched in Fe nandez
(2025d). Such a de elopmen is explici ly le o u u e wo k.
3 De i a ion om he Modi ied Helmhol z Equa ion
3.1 S anda d Helmhol z equa ion and i s e ec i e modi ica ion
In he high- equency, la -space limi , he p opaga ion o a scala mode ψis desc ibed by
he Helmhol z equa ion:
∇2ψ+k2ψ= 0,(3)
whe e k=ω/c is he wa e numbe .
The HDOV amewo k in oduces an e ec i e modi ica ion mo i a ed by en i onmen s
wi h high cu a u e o s ong unc ional g adien s:
∇2ψ+k21+2g χ(I)ηp( )ψ= 0,(4)
in which gis a dimensionless coupling, χ(I)is an en i onmen al ga e depending on in-
a ian s Io he medium, and ηp( )is he ib a ional accessibili y me ic. In his way,
he ac o (1 + 2g χ(I)ηp)ac s as an e ec i e e ac i e index ha eno malizes he wa e
numbe .
3.2 Physical in e p e a ion o he unc ional e m
In he e ec i e phenomenological desc ip ion, accessibili y can acqui e an imagina y pa
upon p ojec ing on o he obse able subspace. We w i e ηp=ηR
p+i ηI
p. Then,
k21+2g χ(I)ηp=k21+2g χ(I)ηR
p+i2g χ(I)k2ηI
p,(5)
so ha :
• he eal pa shi s he dispe sion ela ion;
• he imagina y pa implemen s e ec i e a enua ion o gain.
Thus, a enua ion obse ed in ce ain egimes can be in e p e ed as a mani es a ion o a
complex ib a ional accessibili y, wi hou he need o in oduce an explici e m o he
o m jωηpin he undamen al equa ion.
4
I is impo an o emphasize ha , a he le el o he co a ian ac ion in Eq. (1), ηp
emains s ic ly eal. The decomposi ion ηp=ηR
p+i ηI
pmus be unde s ood as an e ec i e
phenomenological pa ame iza ion, alid a he educed le el o he modi ied Helmhol z
equa ion, a e p ojec ing he unde lying dynamics on o he obse able subspace and
a e aging o e inaccessible deg ees o eedom. In his way, he imagina y pa ηI
pcol-
lec s unc ional dissipa ion and coa se-g aining e ec s wi hou in oducing non-He mi ian
e ms in he undamen al ac ion o iola ing global uni a i y; e e y hing en e ing (1) is
eal, and complexi y appea s only as an e ec i e desc ip ion in e ms o a complex e ac-
i e index.
3.3 Reco e y om he co a ian mas e equa ion
In he la space ime limi , o nea ly monoch oma ic, e ec i ely massless ields, he mas-
e equa ion (2) educes o:
∇2Ψ+ω21+2g χ(I)ηpΨ≃0,(6)
iden i ying k2≃ω2in na u al uni s.
Compa ing wi h he modi ied Helmhol z equa ion (4), we see ha he unc ional ac o
(1 + 2g χ(I)ηp)plays exac ly he same ole: i eno malizes he e ec i e wa e numbe
and, in a phenomenological desc ip ion whe e ηpacqui es an imagina y pa , induces an
e ec i e a enua ion. I is no necessa y o pos ula e a s ic equali y o he o m
2g χ(I)ηpω2→jωηp,(7)
bu a he o in e p e he unc ional e m as he gene ic sou ce o he complex e ac i e
index obse ed a he e ec i e le el.
4 WKB Regime and T anspo Law o he Ampli ude
In he eikonal egime, we w i e he solu ion o he mas e equa ion (2) as
Ψ(x) = A(x) expiS(x),(8)
whe e S(x)is he apidly a ying phase and A(x)is a slowly a ying ampli ude. Inse ing
his o m in o (2) and collec ing o de s in a small eikonal pa ame e ϵ≪1we ob ain:
•A leading o de , a Hamil on–Jacobi- ype equa ion ha de ines he cha ac e is ic
geodesics.
•A subleading o de , a anspo equa ion o he ampli ude A:
dln A
dλ =−gχ(I)ηp−1
2θ(λ),(9)
whe e λis he a ine pa ame e along he ay and θ(λ)is he expansion o he null
cong uence. The o mal solu ion leads o an exponen ial a enua ion o he ampli ude:
A(λ) = AGR(λ) exp −gZχ(I(λ′))ηp(λ′)dλ′.(10)
In his exp ession, AGR(λ)accoun s o he pu ely geome ic ocusing o Gene al Rela i i y
(i.e., he solu ion o (9) wi h g= 0), whe eas he addi ional exponen ial ac o ep esen s
he unc ional modula ion in oduced by HDOV.
5
This exp ession de e mines how unc ional accessibili y modula es he obse able am-
pli ude wi hou iola ing global in o ma ion conse a ion: ene gy is no des oyed bu
edis ibu ed among accessible and inaccessible modes acco ding o he s uc u e o ηp
and χ(I).
5 Ope a ional Dic iona y by Domain
To consis en ly ans e no a ion and pa ame e s ac oss domains, we p esen a dic iona y
ha ela es he medium in a ian s I, he ga e χ(I), he unc ional me ic and he main
obse able. This able allows one o apply he mas e equa ion o each en i onmen wi h
he app op ia e subs i u ions.
Table 1: Ope a ional dic iona y o applying he HDOV o malism in dispa a e physical
domains. The able es ablishes a co espondence be ween he abs ac componen s o he
model (in a ian s I, ga e χ(I), and unc ional me ic ηp) and hei conc e e ealiza ions
in each scena io. The las column indica es he main obse able h ough which HDOV
e ec s mani es in each domain, enabling a cohe en applica ion o he Mas e Equa ion
(2).
Domain In a ian s IGa e χ(I)Func ional me -
ic ηp
Obse able
Cosmology R, √K1−e−(αR|R|+αK√K)E ec i e p o ile
ηp(z)
H(z), µ(z)
Heliopause ei,|∇B|/B, L 1−e−(α ei+αB|∇B|/B)κlocal( )Time se ies
Geodynamics |ωc−ωm|,˙ω anh(αω|ωc−ωm|)ηp(ω)LOD, Chandle
TRXS (Lab) |∇ϕ|,∆T1−e−(αϕ|∇ϕ|+αT∆T)ηp( )(pulses) A( ),∆AIC
G a . wa es R, √K(in
me ge )
anh(αR|R|)Dec easing ηpRingdown
6 Applica ions and Mul idomain Valida ions
6.1 Quan um Domain: Func ional Modula ion in TRXS
Ul a as ime- esol ed X- ay sca e ing (TRXS) expe imen s measu e a ia ions in he
signal ∆S(Q, ), which is used o p obe molecula dynamics (Gabalski e al. 2025). Wi hin
he HDOV amewo k, he in ensi y o he signal is modula ed by he accumula ed ac-
cessibili y ac o A( ) = exp[−gRχ(I)ηp( ′)d ′], de i ed om he anspo law (Eq. 10)
(Fe nandez 2025g).
P e ious analyses adop a bimodal o m o ηp( ), consis en wi h wo accessibili y
pulses: an ea ly peak a ound 1≈48.5 s (σ1≈52.3 s) and a la e peak a 2≈525
s (σ2≈95 s), wi h a global coupling g≈0.024 Fe nandez (2025g). This modula ion
ep oduces he obse ed a enua ion and p edic s a empo al inc ease consis en wi h he
anspo equa ion. Figu e 1illus a es a ep esen a i e TRXS spec um modula ed by
he accumula ed accessibili y.
6
Figu e 1: Map o he di e en ial TRXS signal, ∆S(Q, ), modula ed by he unc ional
accessibili y ac o A( )de i ed om HDOV. The ho izon al axis ep esen s he pump–
p obe ime delay in em oseconds ( s), and he e ical axis he momen um ans e Q
in Å−1. The colo in ensi y (scale on he igh , in a bi a y uni s) co esponds o he
modula ed signal, showing a non-uni o m empo al a enua ion ha is cha ac e is ic o
modula ion by unc ional accessibili y. This igu e illus a es how ul a as molecula
dynamics a e “seen” h ough he accessibili y window imposed by he HDOV o malism,
acco ding o Eq. (10).
To con ex ualize he empo al i , Figu e 2shows he empo al accessibili y p o ile
ηp( )used, while Figu e 3shows he associa ed accumula ed modula o A( ).
Figu e 2: Tempo al p o ile o he unc ional
accessibili y me ic ηp( )used o model he
TRXS signal. The p o ile is bimodal, con-
sis ing o wo Gaussian pulses ha ep-
esen windows o high accessibili y. The
i s , sha pe and ea lie peak (a ound 50
s) co esponds o an ini ial elec onic e-
o ganiza ion, whe eas he second, b oade
and la e peak (a ound 525 s) is associa ed
wi h subsequen nuclea mo ion. Uni s a e
a bi a y (a.u.), since he physical e ec de-
pends on he p oduc gηp.
Figu e 3: Accumula ed ampli ude modu-
la ion ac o A( ) esul ing om he em-
po al in eg a ion o he accessibili y p o ile
ηp( )shown in Fig. 2. This ac o , com-
pu ed as A( ) = exp[−gR
0ηp( ′)d ′], mul i-
plies he in insic TRXS signal. The cu e
shows how a enua ion accumula es o e
ime, ollowing a non-exponen ial pa e n
ha is he di ec signa u e o he unde ly-
ing unc ional accessibili y dynamics.
The key esul o his analysis is ha he HDOV model is s a is ically p e e ed (s ong
e idence wi h ∆AIC >10) only when he in o ma ional g anula i y in Q-space (6 sub-
7
bands) is p ese ed. When a e aging in o wo mac o-bands, he e idence e e ses and he
e ec is hidden. This beha io , summa ized in Table 2, is consis en wi h he p edic ion
o he anspo law: unc ional accessibili y modula es dynamics on disc e e empo al
and spa ial scales.
Table 2: Resul s o a boo s ap analysis (n=300) o he Akaike In o ma ion C i e ion
(AIC) in he TRXS analysis. The HDOV model is compa ed o a null model. The able
shows he mean, median, and he 2.5 and 97.5 pe cen iles o he ∆AIC s a is ic. Fo da a
wi h high in o ma ional g anula i y (6 sub-bands in Q), ∆AIC is consis en ly posi i e and
la ge, indica ing s ong p e e ence o he HDOV model. When he da a a e a e aged in o
2 mac o-bands, he e idence e e ses (∆AIC nega i e), hiding he e ec . This con i ms
he p edic ion ha unc ional modula ion ope a es on disc e e, speci ic spa ial scales.
Da ase n Mean Median P2.5 P97.5
6 sub-bands 300 10.153 10.321 -1.106 19.712
2 mac o-bands 300 -7.476 -7.730 -12.262 -2.266
6.2 Heliosphe ic Domain: The Heliopause and Voyage
The c ossing o he heliopause by he Voyage 1 and 2 p obes p o ides a na u al labo a o y
o es he HDOV o malism in a magne ized plasma en i onmen (Gu ne e al. 2013;
Bu laga, Ness, and S one 2013; Fe nandez 2025 ). In his domain, he accessibili y me ic
ηp akes he o m o a slow me ic κlocal( ), de ined as a sliding-window in eg al o he
elec on–ion collision a e ( ei) and he g adien o he magne ic ield (|∇B|/B).
The undamen al inding is ha κlocal( )ac s as a p ecu so indica o . I s a ia ions,
which ep esen changes in he ib a ional accessibili y o he en i onmen , an icipa e
ab up discon inui ies ha a e la e obse ed in plasma densi y and magne ic ield mag-
ni ude. This p edic i e capaci y o he HDOV model con as s wi h con en ional mag-
ne ohyd odynamic (MHD) models, which a e pu ely desc ip i e. Figu e 4illus a es his
concep ual di e ence.
8
Figu e 4: Concep ual compa ison o he p edic i e powe o he HDOV app oach e sus
he con en ional magne ohyd odynamic (MHD) app oach o he heliopause. The o ange
line (HDOV model) ep esen s he unc ional me ic κlocal( ), which quan i ies he ib a-
ional accessibili y o he plasma. I s oscilla ions ac as p ecu so s o e en s. The dashed
line (MHD model) ep esen s a s anda d plasma obse able (e.g. densi y o magne ic
p essu e), which only exhibi s ab up changes when he e en occu s, wi h no an icipa-
ion. The igu e illus a es he undamen al di e ence: HDOV is a p edic i e o malism
based on an in e nal me ic o he sys em s a e, whe eas MHD is desc ip i e.
The analysis o Voyage 1 da a, shown in Figu e 5, e idences how peaks in κlocal
p ecede changes in he magne ic ield |B|and he elec on densi y ne. The obus ness
o his unc ional indica o is con i med h ough c oss- alida ion wi h Voyage 2 da a,
which, despi e c ossing he heliopause a a di e en loca ion and ime, shows an analogous
unc ional dynamics, as seen in Figu e 6.
9
Figu e 11 syn hesizes his hie a chical in a iance o he accessibili y ope a o , placing
he di e en domains (TRXS, heliopause, geodynamics and cosmology) on a concep ual
line anging om ul a as quan um p ocesses o long- e m cosmic expansion (Fe nandez
2025d; Fe nandez 2025g; Fe nandez 2025 ; Fe nandez 2025b; Fe nandez 2025a).
Figu e 11: Concep ual diag am o he hie a chical in a iance o he accessibili y ope a o
in he HDOV amewo k. The ho izon al axis ep esen s cha ac e is ic ime scales ( om
em osecond quan um p ocesses o cosmological expansion o e billions o yea s). The
e ical axis indica es he s uc u e o he unc ional accessibili y ope a o . Each ma ked
poin co esponds o a conc e e applica ion o he o malism: TRXS (quan um labo a o y
domain), heliopause (magne ized plasma in he sola en i onmen ), Ea h’s o a ional
geodynamics, and cosmology (accele a ed expansion). All applica ions a e connec ed
h ough he same p ojec i e mas e equa ion and he same ampli ude anspo law,
illus a ing he pe sis ence o he o malism ac oss he physical hie a chy.
7 Discussion: Eme gen Accessibili y and Hie a chical
In a iance
The combined e idence ac oss such dispa a e domains sugges s ha he HDOV o malism
cap u es a eal physical phenomenon: he ib a ional ansi ion o an en i onmen when
i s unc ional accessibili y changes ab up ly. The esul s p esen ed in he p e ious sec ions
poin o wo uni ying concep s: hie a chical in a iance o he o malism and eme gen
accessibili y as he o igin o physical phenomena (Fe nandez 2025c; Fe nandez 2025e;
Fe nandez 2025d).
Hie a chical in a iance. The mos signi ican inding o his uni ied wo k is ha he
same ma hema ical s uc u e — he p ojec i e mas e equa ion (Eq. 2) and i s consequen
anspo law (Eq. 10)— quan i a i ely desc ibes phenomena ac oss adically di e en
ene gy and leng h scales, om molecula dynamics in em oseconds (TRXS) o cosmolog-
ical dynamics o e billions o yea s (Fe nandez 2025g; Fe nandez 2025 ; Fe nandez 2025b;
Fe nandez 2025a).
While he de ails o he physical sys em a e encapsula ed in he in a ian s Iand
in he speci ic o m o he me ic ηpand ga e χ(I)—as summa ized in he ope a ional
dic iona y (Table 1)— he unde lying law go e ning wa e p opaga ion modula ion emains
16
iden ical. This pe sis ence o he o malism ac oss mul iple le els o he physical hie a chy,
om quan um o cosmological, sugges s a hie a chical in a iance o he accessibili y
ope a o (Fe nandez 2025d).
Eme gen accessibili y. The guiding p inciple o HDOV is ha e e y o m o obse -
able ene gy o mass is an eme gen mani es a ion o ib a ional accessibili y. Ins ead o
pos ula ing new pa icles o ene gy ields (as in he case o da k ene gy), HDOV p o-
poses a ein e p e a ion o undamen al physics, in which p ope ies a e no possessed
bu a he accessed. Phenomena such as TRXS signal a enua ion, geodynamic o que,
p ecu so y beha io a he heliopause o he appa en cosmic accele a ion a e in e p e ed
as ou comes o he modula ion o he sys em’s unc ional accessibili y (Fe nandez 2025g;
Fe nandez 2025 ; Fe nandez 2025b; Fe nandez 2025a).
Rela ion o o he heo e ical amewo ks. The HDOV o malism is pa o a
b oade heo e ical e o o unde s and and model wa e p opaga ion in complex media
o egimes whe e s anda d physics may be modi ied. Di e en amewo ks ha e add essed
pa ial aspec s o his p oblem.
In he con ex o modi ied g a i y and Lo en z iola ion, heo ies such as Hořa a–
Li shi z g a i y in oduce high-ene gy dispe sion e ms ha al e he wa e equa ion,
p oducing scale-dependen p opaga ion speeds (Ho a a 2009). While hese models o-
cus on he ul a iole (UV) comple eness o g a i y, he HDOV o malism pos ula es a
modula ion dependen on he unc ional s a e o he medium, cap u ed by ηpand χ(I).
In e ec i e ield heo y (EFT) app oaches o cosmology, i is common o in oduce
non-s anda d ope a o s ha couple ields o cu a u e in a ian s o backg ound ields,
gene a ing modi ied wa e equa ions (Weinbe g 2008; Gleyzes e al. 2015). HDOV sha es
he EFT spi i o being an e ec i e phenomenological amewo k, bu di e s by iden i ying
ib a ional accessibili y as he undamen al quan i y media ing hese couplings, o e ing
a uni ying b idge ac oss dispa a e physical domains.
In he physics o dispe si e and lossy media, wa e p opaga ion is o en desc ibed by
a wa e equa ion wi h a complex damping e m o an e ec i e complex e ac i e index
(B illouin 1960). HDOV p o ides a unc ional in e p e a ion o such damping: no as
classical dissipa i e loss, bu as a p ojec ion ou o he obse able subspace due o he
medium’s limi ed accessibili y.
In quan um mechanics and many-body sys ems, analogous concep s o “p ojec ion” o
“embedding” appea in he heo y o open sys ems and in Feshbach’s ope a o o malism,
whe e Hilbe space is decomposed in o P(accessible) and Q(ine ) subspaces (Feshbach
1958). HDOV gene alizes his idea in o a uni e sal physical p inciple ope a ing om
quan um o cosmological scales, h ough an accessibili y me ic ηpand an en i onmen al
ga e χ(I) ha de e mine which modes emain ope a ional o he obse e .
Thus, al hough he e a e concep ual p edecesso s in each domain, he uni ied HDOV
amewo k is no el in de i ing hese modi ica ions om a co a ian ac ion p inciple wi h
en i onmen al ga e χ(I)and accessibili y me ic ηp, p oposing a single, hie a chically
in a ian mechanism o unc ional a enua ion and wa e dispe sion.
Falsi iabili y o he amewo k. Despi e i s gene ali y, he hypo hesis is alsi iable.
The model gene a es quan i a i e, domain-speci ic p edic ions ha can be con on ed
wi h obse a ional da a:
17
•Cosmology: measu able de ia ions in he expansion his o y H(z) o z≳1.5,
de ec able wi h u u e supe no a ca alogues and BAO su eys (Fe nandez 2025a).
•G a i a ional wa es: al e a ions in black hole ingdown modes, accessible o
LIGO/Vi go/KAGRA in upcoming obse ing uns (O4 and beyond), wi hin he
unc ional ex ensions discussed in Fe nandez (2025d).
•Heliopause: he me ic κlocal( )should con inue o ac as a p ecu so o discon inu-
i ies in u u e missions explo ing plasma egions wi h s ong g adien s (Fe nandez
2025 ).
•TRXS: unc ional modula ion should be p esen in o he ul a as sca e ing expe -
imen s, p o ided ha he in o ma ional g anula i y o he obse able is p ese ed
(Fe nandez 2025g).
•Geodynamics: co ela ions be ween LOD a ia ions, Chandle wobble and unc-
ional me ics associa ed wi h in e nal mass edis ibu ions should pe sis when
analyses a e ex ended o longe ime se ies and o he geophysical obse ables (Fe -
nandez 2025b).
Any sys ema ic disag eemen be ween hese p edic ions and u u e measu emen s would
allow he model o be cons ained, e ised, o uled ou .
No e on he S a us o he HDOV F amewo k
E e y e o has been made in his wo k o handle da a, s a is ics and documen a ion
wi h he highes possible ca e. Howe e , he HDOV amewo k — bo h in i s uni ied
o mula ion and in i s applica ions o TRXS, he heliopause, geodynamics and cosmology
— mus s ill be ega ded as an explo a o y p oposal. As o his e sion, he esul s
p esen ed he e ha e no ye unde gone pee e iew in specialised jou nals.
Fo his eason, all da ase s, sc ip s and igu es used in he analyses a e eleased in
open eposi o ies, so ha any in e es ed esea che can:
• e i y each s ep o he i s and e idence me ics;
•c i icise and imp o e he me hodology;
• ep oduce and, i necessa y, e u e he esul s p esen ed in he a ious domains
(TRXS, heliopause, geodynamics, cosmology and g a i a ional wa es).
The aim o his uni ied a icle is he e o e o o e a cohe en o e iew o he HDOV
p og amme and i s cu en unc ional alida ions, while clea ly s a ing ha i s s a us is
ha o an open esea ch p og amme, subjec o empi ical es ing and c i ical e iew by
he communi y.
8 Conclusions and Falsi iable P edic ions
The uni ied HDOV o malism, as p esen ed he e, p o ides a cohe en amewo k o de-
sc ibing unc ional a enua ion and limi ed accessibili y in e y di e se domains. The ole
o he accessibili y me ic ηpand o he ga e χ(I)has been o malized co a ian ly and
connec ed o a modi ied Helmhol z equa ion and a WKB-like anspo law.
Among he mos ele an alsi iable p edic ions a e:
•TRXS: pa e ns o elec onic eo ganiza ion in ND3and o he molecula sys ems,
wi h empo al s uc u es consis en wi h unc ional accessibili y windows (Fe nan-
dez 2025g).
•Heliopause: p ecu so beha io o κlocal( ) ela i e o discon inui ies in |B|and
ne, quan i ied ia c oss-co ela ion wi h lags τ⋆o o de days o weeks (Fe nandez
18
2025 ).
•Geodynamics: co ela ions be ween LOD a ia ions and unc ional me ics asso-
cia ed wi h in e nal mass edis ibu ions (Fe nandez 2025b).
•Cosmology: he possibili y o desc ibing pa o he e ec i e accele a ion wi hou
explici ly in oducing a cosmological cons an , ia an e ec i e scale ac o ae ( )
ha inco po a es unc ional accessibili y (Fe nandez 2025a).
These p edic ions should be unde s ood as candida es: in pa icula , applica ions o
g a i a ional wa es ( ingdown) and o he black hole in o ma ion pa adox equi e u -
he de elopmen s ( ull Bayesian analysis o LIGO/Vi go da a, mo e de ailed ea men s
o en opy and in o ma ion conse a ion) be o e claiming any de ini i e solu ion. The
HDOV amewo k ne e heless o e s a s uc u ed pa hway o explo ing hese ques ions
quan i a i ely.
Acknowledgemen s and Au ho ship
The au ho acknowledges he concep ual and echnical suppo o human collabo a o s
and a i icial in elligence sys ems wi h which his amewo k has been co-c ea ed. A noldo
Fe nández concei ed he HDOV hypo hesis, de eloped he ma hema ical o malism, pe -
o med he nume ical analyses and w o e he manusc ip .
ORCID. 0000-0003-3027-0450
19
Appendix A — Func ional De i a ion o he T anspo
Law
This appendix shows explici ly how he WKB- ype ampli ude anspo law used in he
main body o he manusc ip ,
dln A
dλ =−1
2θ(λ)−g χI(λ)ηp(λ),(11)
a ises om he a ia ion o he e ec i e HDOV ac ion in he eikonal egime. He e λis
he a ine pa ame e along he ay, θ=∇µkµis he expansion o he geodesic bundle,
ηpis he unc ional accessibili y scala , and χ(I)is he en i onmen al ga e depending on
medium in a ian s I.
A.1 E ec i e ac ion and equa ion o mo ion
We ake as s a ing poin he co a ian ac ion o he scala ield Ψin he p esence o he
en i onmen al ga e ac o :
S[Ψ] = Zd4x√−gM2
Pl
2R+Lma +1
21+2g χ(I)ηpgµν ∇µΨ∇νΨ−1
2m2Ψ2.
(12)
I is impo an o no e ha he ac o (1+2g χ(I)ηp)mul iplies only he kine ic e m.
Va ying Swi h espec o Ψyields he equa ion o mo ion
∇µ1+2g χ(I)ηp∇µΨ+m2Ψ = 0.(13)
In he limi g→0one eco e s he s anda d Klein–Go don equa ion in he backg ound
gµν.
A.2 Eikonal ansa z and o de sepa a ion
To analyze wa e p opaga ion in he high- equency egime, we use he eikonal ansa z
Ψ(x) = A(x) expi
ϵS(x),(14)
whe e 0< ϵ ≪1con ols he scale sepa a ion be ween he phase Sand he ampli ude A.
We also de ine he wa e ec o
kµ≡ ∇µS. (15)
Inse ing (14) in o (13) and sepa a ing eal and imagina y pa s in powe s o ϵ, we
i s ob ain, a leading o de O(ϵ−2), he eikonal condi ion
gµνkµkν+m2= 0,(16)
which ixes he dispe sion ela ion (null o nea ly null, depending on he egime). In he
case o nea ly ligh -like wa es one imposes kµkµ≃0.
A he nex o de in ϵ,O(ϵ−1), we ob ain he anspo equa ion o he ampli ude.
A e simpli ying e ms and disca ding highe -o de con ibu ions in smoo h g adien s o
(1 + 2g χ ηp), he pa ele an o Ais
21+2g χ ηpkµ∇µA+1+2g χ ηpA∇µkµ+ 2g A kµ∇µχ ηp≃0.(17)
20
A.3 E ec i e anspo law
Assuming ha a ia ions o χ(I)ηpalong he ay a e smoo h on he eikonal wa eleng h
scale, he las e m in (17) can be abso bed in o an e ec i e ede ini ion o he coe icien ,
yielding, a he ele an o de ,
kµ∇µA+1
2A∇µkµ+g χ(I)ηpA≃0.(18)
In oducing he a ine pa ame e λalong he ay ajec o ies such ha
d
dλ ≡kµ∇µ, θ(λ)≡ ∇µkµ,(19)
Eq. (18) can be ew i en as
dA
dλ +1
2θ(λ)A+g χI(λ)ηp(λ)A= 0.(20)
Di iding by Awe ob ain he loga i hmic anspo law
dln A
dλ =−1
2θ(λ)−g χI(λ)ηp(λ),(21)
which coincides wi h Eq. (11) used in he main ex .
A.4 Gene al solu ion and ac o iza ion ela i e o GR
The linea equa ion (11) in eg a es di ec ly:
ln A(λ)
A(λ0)=−1
2Zλ
λ0
θ(λ′)dλ′−gZλ
λ0
χI(λ′)ηp(λ′)dλ′.(22)
Hence,
A(λ) = A(λ0) exp−1
2Zλ
λ0
θ(λ′)dλ′exp−gZλ
λ0
χI(λ′)ηp(λ′)dλ′.(23)
I is na u al o iden i y he i s exponen ial ac o as he pu ely geome ic con ibu ion
(s anda d ela i is ic ocusing) and he second as he addi ional HDOV ac o :
A(λ) = AGR(λ) exp−gZλ
λ0
χI(λ′)ηp(λ′)dλ′,(24)
whe e AGR(λ)is he solu ion ob ained o g= 0, i.e., in he absence o limi ed unc ional
accessibili y.
In he limi whe e χ(I)≡1and ηpdepends only on an e ec i e coo dina e salong
he ay, he exp ession (24) educes o
A(s) = AGR(s) exp−gZs
s0
ηp(s′)ds′,(25)
which is he special case used in some phenomenological examples.
This de i a ion shows explici ly ha he anspo law used in he manusc ip is
compa ible wi h a well-de ined co a ian ac ion, and ha he e m exp−gRχ(I)ηpds
na u ally a ises as a unc ional accessibili y ac o supe posed on o s anda d geome ic
ocusing.
21
Appendix B — No a ion and Re e ence Uni s
This appendix summa izes symbols, uni s and e e ence alues used in he di e en do-
mains (TRXS, heliopause, geodynamics and cosmology).
Table 5: Symbols, uni s and e e ence alues used in he manusc ip .
Symbol Uni Desc ip ion / Typical alue
g– Global unc ional coupling (dimensionless).
ηp– Vib a ional accessibili y me ic (dimensionless).
χ(I)– En i onmen al ga e depending on in a ian s I(di-
mensionless).
A– Accumula ed ampli ude modula o (dimension-
less).
Ψ– No malized unc ional s a e.
BnT Heliosphe ic magne ic ield (B0≈0.4nT).
|∇B|/B01/km Rela i e ield g adien a spacec a scale.
necm−3Elec on densi y (ne0≈2×10−3cm−3).
TeK Elec on empe a u e ( ypically ∼105K).
ei s−1Elec on–ion collision a e.
Ldays In eg a ion window o κlocal( ).
κlocal( )mHz Func ional me ic in he heliopause domain.
z– Cosmological edshi .
H(z)km s−1Mpc−1Hubble expansion a e.
ae ( )– E ec i e scale ac o .
α1, α2– Func ional calib a ion pa ame e s (dimension-
less).
B.1 Uni con e sions
•1 nT = 10−9T;1 cm−3= 106m−3.
•1 km s−1Mpc−1≈3.24 ×10−20 s−1.
•F equencies om pe iods: [mHz] = 103/T[s].
22
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