Unified Matrix--QCA Universe Theory of Gravitational Wave Lorentz Violation and Dispersion\\ \large Bounds on $v_{\mathrm g

Unied Ma ixQCA Uni e se Theo y o
G a i a ional Wa e Lo en z Viola ion and Dispe sion
Bounds on
g=c
and Tes able P edic ions unde Unied Time Scale
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 19, 2025
Abs ac
Unde he unied amewo k o unied ime scale, bounda y ime geome y, Ma-
ix Uni e se THEMATRIX, and Quan um Cellula Au oma on (QCA) Uni e se,
we cons uc a s uc u al heo y specically o G a i a ional Wa e Lo en z Vio-
la ion and Dispe sion Co ec ions. The unied ime scale is dened by he scale
iden i y o sca e ingspec al shi Wigne Smi h g oup delay
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω), Q(ω) = −iS(ω)†∂ωS(ω),
(1)
uni ying sca e ing hemi-phase de i a i e, ela i e densi y o s a es, and g oup delay
ace in o a single ime densi y
κ(ω)
, whose in eg al denes he ime scale equi -
alence class ep esen a i e
τsca (ω)
. In he pe spec i e o g a i a ional wa es as
sca e ing modes o geome ic pe u ba ions,
κ(ω)
di ec ly con ols he phase e-
loci y, g oup eloci y, and equency-dependen p opaga ion delay o g a i a ional
wa es.
In he Uni e se QCA objec
UQCA = (Λ,Hcell,Aqloc, α, ω0),
(2)
g a i a ional deg ees o eedom a e embedded as linea ized exci a ions o g a i y
QCA modes, whose quasi-ene gy spec um
ε(k)
yields an eec i e dispe sion ela-
ion in he con inuous limi
ω2=c2k21 + ε2(kℓcell)2+ε4(kℓcell)4+· · · ,
(3)
whe e
ℓcell
is he QCA eec i e la ice spacing and
ε2n
a e dimensionless coecien s.
The unied ime scale equi es he QCA disc e e ime s ep o be in he same equi -
alence class as geome ic p ope ime, bounda y modula ime, and sca e ing ime
scale, he eby di ec ly linking
ε2n
in g a i a ional wa e dispe sion o high-o de
de ia ions o
κ(ω)
.
Unde app op ia e spec alsca e ing and QCA axioms, his pape ob ains he
ollowing main esul s:
(1) In he Ma ix Uni e se ep esen a ion, iewing weak-eld g a i a ional wa es
as linea pe u ba ion modes on backg ound FRW/a space ime, we cons uc he
g a i a ional wa e sca e ing ma ix
SGW(ω;k)
and g oup delay ma ix
QGW(ω;k)
,
1
p o ing ha in he a -eld low- equency limi , he de ia ion o he unied scale
densi y
δκGW(ω)
and he dispe sion unc ion
ε(k)
sa is y
δ g(ω) = ∂ω
∂k −c≃chε2(kℓcell)2+O(kℓcell)4i, δκGW(ω)∼ − L
2πc2δ g(ω),
(4)
whe e
L
is he eec i e p opaga ion dis ance.
(2) We cons uc a class o G a i yQCA Models in he QCA Uni e se, whose
linea ized deg ees o eedom ep oduce he ans e se aceless g a i a ional wa e
equa ion o Gene al Rela i i y (GR) in he long-wa e limi , while high-o de
(kℓcell)2n
dispe sion e ms a e de e mined by cellula s uc u e and upda e ules. Unde uni-
ed ime scale and bounda y ime geome y cons ain s, combining disc e e sym-
me ies and NullModula double co e consis ency, we p o e ha in he absence
o chi al anomalies and wi h ime e e sal conse a ion, g a i a ional wa e dispe -
sion only allows e en-o de
(kℓcell)2n
ype co ec ions, while odd-o de
k2n+1
ype
Lo en z iola ions a e excluded in he unied amewo k.
(3) U ilizing cons ain s on g a i a ional wa e p opaga ion speed and dispe -
sion om LIGOVi goKAGRA and he mul i-messenge e en GW170817/GRB
170817A (e.g., speed cons ain
| g/c −1|≲10−15
and mul i-e en  s o pa am-
e e ized dispe sion ela ions in GWTC ca alogs), we ew i e hese esul s in he
unied amewo k as uppe bounds on QCA la ice spacing
ℓcell
and dispe sion co-
ecien s
ε2, ε4
. Combining exis ing cons ain s on ene gy scale
M∗
o
n= 2
ype
k4
co ec ions, we ob ain
ℓcell ≲M−1
∗|β2|−1/2, β2=O(1),
(5)
whe e
M∗
ypically lies in he
1013

1015 GeV
ange, co esponding o
ℓcell ≲10−29

10−31 m
.
This esul is o compa able magni ude o independen cons ain s on disc e e space-
ime and QCA la ice spacing based on elec omagne ic and ma e in e e ome y
expe imen s.
(4) In he unied causalen opy ime amewo k, we p o e a G a i yQCA
Causal Consis ency Theo em: i he eec i e ligh cone o G a i yQCA emains
consis en wi h he causal ligh cone o bounda y ime geome y in he LIGO/Vi go
equency band, hen allowed Lo en z iola ions mus exhibi specic e en-o de
dispe sion s uc u es, and he g oup eloci y de ia ion sa ises

δ g(ω)
c
≲O(ωℓcell)2
(6)
Planck-scale supp ession law, wi h high-o de con ibu ions o g oup delay being
exponen ially supp essed unde cu en obse a ional p ecision.
(5) The appendix p o ides: cons uc ion om GR linea pe u ba ions o Ma ix
Uni e se sca e ing ma ix
SGW(ω;k)
; con inuum limi and dispe sion expansion o
G a i yQCA models; p ecise ela ionship be ween g oup delay and
κ(ω)
de ia ion
unde unied ime scale; and he p ocess o con e ing LIGO/Vi goGW170817 and
GWTC-3 cons ain s in o nume ical bounds on
(ℓcell, ε2)
.
Resul s indica e: in he Unied Ma ixQCA Uni e se Theo y, g a i a ional
wa e Lo en z iola ion and dispe sion a e no a bi a y high-dimensional ope a o
pe u ba ions, bu geome icspec al p ojec ions o QCA disc e e s uc u e and
unied ime scale de ia ions. Exis ing obse a ions ha e al eady comp essed his
de ia ion o an ex emely small ange, p o iding s ong cons ain s on he la ice
spacing and dispe sion coecien s o he uni e se's disc e e s uc u e, and oe ing
a es able unied empla e o u u e high- equency and mul i-band g a i a ional
wa e de ec ion.
2
Keywo ds:
G a i a ional wa es; Lo en z in a iance iola ion; Dispe sion ela ion; Uni-
ed ime scale; Sca e ing ma ix; Wigne Smi h g oup delay; Quan um cellula au-
oma a; Ma ix uni e se; S anda d-Model Ex ension; GW170817; GWTC-3
1 In oduc ion & His o ical Con ex
1.1 G a i a ional Wa e P opaga ion and Lo en z In a iance
In Gene al Rela i i y (GR), weak-eld g a i a ional wa es a e ans e se aceless enso
pe u ba ions p opaga ing on a Lo en zian backg ound geome y, sa is ying he linea ized
Eins ein equa ions wi h dispe sion ela ion
ω2=c2k2
, whe e phase eloci y and g oup
eloci y a e bo h equal o he speed o ligh
c
. Any phenomenon de ia ing om his
dispe sion ela ion can be ega ded as Lo en z in a iance iola ion in g a i a ional wa e
p opaga ion o eec i e medium co ec ion. In eec i e eld heo y language, such
co ec ions a e ypically w i en as
ω2=c2k2+αdispk2+n,
(7)
o equi alen o ms pa ame e ized by g a i on mass and high-dimensional ope a o s,
whe e
αdisp
and
n
a e de e mined by he specic heo y.
G a i a ional wa e de ec ions by LIGO, Vi go, and KAGRA p o ide di ec means o
es hese co ec ions. Sys ema ic analyses based on pa ame e ized dispe sion ela ions
show ha obse able eec s o Lo en z iola ion on wa e o m phases can be embedded
in o he pa ame e ized pos -Eins einian amewo k and join ly cons ained on mul iple
e en s using s anda d Bayesian in e ence.
1.2 GW170817 and G a i a ional Wa e Speed Cons ain s
The 2017 bina y neu on s a me ge e en GW170817 and i s gamma- ay bu s coun-
e pa GRB 170817A ep esen a miles one mul i-messenge e en in he eld o g a i-
a ional wa es. The a i al ime die ence be ween g a i a ional wa es and gamma ays
was abou
1.74 s
, wi h a p opaga ion dis ance o abou
40 Mpc
, yielding a cons ain on
he ela i e die ence be ween g a i a ional wa e g oup eloci y and he speed o ligh
−3×10−15 ≲ g
c−1≲7×10−16,
(8)
i.e.,
| g/c −1|≲O(10−15)
.
This esul b oadly ules ou a la ge class o da k ene gymodied g a i y models
ha adjus g a i a ional wa e speed on cosmological scales, p o iding s ong cons ain s
on Ho ndeski, Eins einAe he , bime ic heo ies, e c. Fu he mo e, analyses o g a i on
mass and Lo en z iola ion indica e ha cu en LIGO/Vi go da a cons ain he g a i on
mass o he ange
mg≲10−23

10−22 eV
, co esponding o a Comp on wa eleng h g ea e
han
1013 km
.
1.3 GWTC-3, SME, and Pa ame e ized Dispe sion Tes s
Wi h he elease o mul iple ba ches o e en s in GWTC-1/2/3, he LIGOVi goKAGRA
collabo a ion has pe o med sys ema ic p opaga ion es s o Gene al Rela i i y, includ-
ing dimensions o speed, dispe sion, a enua ion, and pola iza ion. A signican class
3
o wo k in ol es gene alized dispe sion ela ions wi h aniso opic, bi e ingen , and dis-
pe si e co ec ions de i ed om Lo en z- iola ing ope a o s in he g a i y sec o o he
S anda d-Model Ex ension (SME). Join cons ain s on coecien s o
d= 5,6
dimen-
sional ope a o s using 90 high-condence e en s in GWTC-3 ha e e ealed no signican
signs o Lo en z iola ion.
In he non-dispe si e limi o p opaga ion speed, independen analyses using a i al
ime delays ac oss mul iple de ec o s ha e also gi en condence in e als o
g
wi hin
he
0.97c

1.01c
ange, and cons ained non-bi e ingen , non-dispe si e Lo en z iola ion
coecien s unde he SME amewo k.
O e all, g a i a ional wa e da a indica e ha in he
10

103Hz
equency band, he
p opaga ion o g a i a ional wa es almos pe ec ly obeys Lo en z in a iance, and any
obse able dispe sion o speed de ia ion mus be ex emely weak.
1.4 Disc e e Space ime, QCA, and G a i a ional Wa e Dispe -
sion
On he o he hand, disc e e space ime and Quan um Cellula Au oma a (QCA) ame-
wo ks p o ide a way o uni y he desc ip ion o how con inuous Lo en z symme y
eme ges om deepe disc e e s uc u es. In quan um walks and QCA models, ni e
la ice spacing
ℓcell
and disc e e ime s ep
∆
de e mine he eec i e dispe sion ela ion,
whose con inuous limi ypically ep oduces Di ac/Weyl/Maxwell equa ions, wi h sligh
iola ions o Lo en z symme y embodied in high-o de
kℓcell
co ec ions.
Recen wo k has a emp ed o use Lo en z iola ion obse a ions om elec omagne ic
spec a and high-ene gy cosmic ays o place uppe bounds on he QCA la ice spacing
ℓcell
. Typical esul s indica e ha
ℓcell
mus be a smalle han cu en ly accessible
expe imen al scales, possibly app oaching he
10−29

10−31 m
ange.
In his con ex , a na u al ques ion a ises: **In a unied Ma ixQCA uni e se, can
g a i a ional wa e Lo en z iola ion and dispe sion be in e p e ed as geome icspec al
p ojec ions o QCA disc e e s uc u e, p ecisely con olled by he unied ime scale den-
si y
κ(ω)
? And how s ong a e he uppe bounds on
ℓcell
and dispe sion coecien s
ε2n
gi en by exis ing LIGO/Vi go/GW170817 cons ain s?**
The pu pose o his pape is o cons uc a unied model, p o ide heo em-based
conclusions, and p opose es able p edic ions cen e ing on his ques ion.
2 Model & Assump ions
2.1 Unied Time Scale Mo he Fo mula and Ma ix Uni e se
The mo he o mula o he unied ime scale is
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω), Q(ω) = −iS(ω)†∂ωS(ω),
(9)
whe e
S(ω)
is he xed-ene gy sca e ing ma ix,
φ(ω) = 1
2a g de S(ω)
is he o al hemi-
phase,
ρ el(ω)
is he ela i e densi y o s a es, and
Q(ω)
is he Wigne Smi h g oup delay
ma ix.
The ime scale pa ame e is dened as
τsca (ω) = Zω
ω0
κ(˜ω) d˜ω,
(10)
4
whe e ane ans o ma ions
τ7→ aτ +b
a e conside ed he same ime scale equi alence
class. P io wo k has p o ed ha unde app op ia e sca e inggeome ymodula ow
axioms, geome ic p ope ime, bounda y modula ime, and sca e ing ime scale belong
o he same equi alence class.
The Ma ix Uni e se THE-MATRIX can be abs ac ed a he spec alsca e ing end
as
Uma =Hchan, S(ω), Q(ω), κ(ω),A∂, ω∂,
(11)
whe e
Hchan
is he channel Hilbe space, and
A∂, ω∂
desc ibe bounda y obse able algeb a
and s a e. The uni e se's causal s uc u e, ime a ow, and gene alized en opy ow a e
gi en by bounda y ime geome y on small causal diamonds.
2.2 Uni e se QCA Objec and G a i a ional Deg ees o F eedom
The Uni e se QCA objec is deno ed as
UQCA = (Λ,Hcell,Aqloc, α, ω0),
(12)
whe e
Λ
is a coun able connec ed g aph (usually
Zd
o i s spa se subg aph),
Hcell
is he
ni e-dimensional cellula Hilbe space,
Aqloc
is he quasilocal
C∗
algeb a on i s inni e
enso p oduc ,
α:Z→Au (Aqloc)
is a
∗
-au omo phism wi h ni e p opaga ion adius
and spa ial homogenei y, and
ω0
is he ini ial uni e se s a e.
G a i a ional deg ees o eedom a e encoded in he ollowing way:
1. Backg ound geome y is encoded as eec i e ligh cone s uc u e: he p opaga ion
adius o
α
and adjacency g aph opology ep oduce he causal cone s uc u e o some
Lo en z mani old
(M, g)
in he con inuous limi ;
2. G a i a ional wa e modes a e linea ized eigenmodes o
U= e−iHe ∆
, whose di -
e ence om backg ound
U0
sa ises he ans e se aceless wa e equa ion o GR in he
low-ene gy limi .
In momen um ep esen a ion, using BlochFloque decomposi ion
U=Z⊕
BZ
U(k) dµ(k),
(13)
whe e
BZ
is he B illouin zone,
U(k)
is uni a y on
Hcell
, wi h spec al decomposi ion
U(k) = X
a
exp−iεa(k)∆ Πa(k),
(14)
whe e
εa(k)
is he quasi-ene gy spec um and
Πa(k)
is he eigenp ojec ion. The g a i a-
ional wa e b anch is deno ed
εGW(k)
, dening eec i e equency
ω(k) = εGW(k)
∆ .
(15)
2.3 Dispe sion Rela ion and QCA La ice Spacing
Le he QCA undamen al la ice spacing be
ℓcell
. In he long-wa e limi
kℓcell ≪1
, Taylo
expansion can be pe o med on
ω(k)
. Assuming he exis ence o a clus e o massless
5

b anches wi h dominan beha io
ω(k)≃ck
(
c
being mac oscopic speed o ligh ), i can
gene ally be w i en as
ω2(k) = c2k2"1 + X
n≥1
β2n(ˆ
k)(kℓcell)2n#,
(16)
whe e
ˆ
k=k/k
is he di ec ion, and
β2n(ˆ
k)
a e dimensionless coecien s. This exp ession
explici ly embodies high-o de co ec ions o dispe sion om QCA disc e e s uc u e.
This pape ocuses on he dominan e m unde iso opic app oxima ion
ω2=c2k21 + β2(kℓcell)2,
(17)
co esponding o he
n= 2
ype pa ame e ized dispe sion
ω2=c2k2+αdispk4
in obse -
a ions.
2.4 Unied Time Scale and G a i a ional Wa e Sca e ing Chan-
nels
Fo a gi en equency
ω
, conside he g a i a ional wa e sca e ing channel subspace
HGW ⊂ Hchan
, co esponding o sca e ing ma ix
SGW(ω)∈U(NGW)
and g oup delay
ma ix
QGW(ω) = −iS†
GW(ω)∂ωSGW(ω),
(18)
whose eigen alues
τj(ω)
a e g oup delays o each channel. The unied ime scale densi y
o he g a i a ional wa e sec o is dened as
κGW(ω) = 1
2π QGW(ω) = 1
2π
NGW
X
j=1
τj(ω).
(19)
In he a -eld app oxima ion, i he p opaga ion dis ance is
L
and assuming all
channels ha e simila g oup eloci y
g(ω)
, hen
¯τ(ω) = 1
NGW X
j
τj(ω)≈L
g(ω),
(20)
hus
κGW(ω)≈NGW
2π
L
g(ω).
(21)
Le ing he e e ence quan i y in GR case be
κ(0)
GW(ω)≈NGWL/(2πc)
, i s de ia ion is
dened as
δκGW(ω) = κGW(ω)−κ(0)
GW(ω).
(22)
3 Main Resul s (Theo ems and Alignmen s)
This sec ion p esen s ou co e heo ems based on he model and assump ions, o malizing
QCA dispe sion and unied ime scale, NullModula consis ency, and he impac o
obse a ional cons ain s on la ice spacing
ℓcell
.
6
3.1 Theo em 3.1 (GW Dispe sion and Unied Time Scale Den-
si y De ia ion)
Theo em 3.1
(G a i a ional Wa e Dispe sion and Unied Time Scale Densi y De i-
a ion)
.
Le g a i a ional wa es p opaga e dis ance
L
in a mac oscopically homogeneous
medium (o cosmic backg ound), wi h eec i e dispe sion ela ion
ω2=c2k21 + ε(k),|ε(k)| ≪ 1,
(23)
and g oup eloci y
g(ω) = ∂ω
∂k >0
(24)
being mono onic in he LIGO/Vi go band. Assume sca e ing ma ix
SGW(ω)
can be
cons uc ed om plane wa e modes, and a -eld Wigne g oup delay
τj(ω)
is equi alen
o pe u ba ion o p opaga ion ime
L/ g(ω)
, hen unied ime scale densi y de ia ion
sa ises
δκGW(ω)≈ − L
2πc2δ g(ω) + Oε2,
(25)
whe e
δ g(ω) = g(ω)−c≃c
2hε(k) + kε′(k)ik=ω/c.
(26)
Specically, when
ε(k) = β2(kℓcell)2+O((kℓcell)4)
,
δ g(ω)≃c β2(kℓcell)2,δ g(ω)
c≃β2(kℓcell)2,
(27)
hus
δκGW(ω)≈ − L
2πc β2(kℓcell)2.
(28)
3.2 Theo em 3.2 (E en-O de Dispe sion S uc u e o QCA G a -
i y Modes)
Theo em 3.2
(E en-O de Dispe sion S uc u e o QCA G a i y Modes)
.
Le
UQCA
be
a spa ially homogeneous, local, ansla ion-in a ian QCA sa is ying he ollowing condi-
ions:
1. Exis ence o pa i y symme y
P
and ime e e sal symme y
T
, such ha
PU(k)P−1=
U(−k)
,
TU(k)T−1=U†(−k)
;
2. Exis ence o a clus e o massless g a i a ional wa e b anches sa is ying
ω(k)∼ck
as
k→0
;
3. This b anch has ni e degene acy wi h o he b anches in he low-ene gy limi , and
can be diagonalized in an app op ia e basis appea ing as pai s o
ω(k)
and
−ω(k)
.
Then he expansion o
ω2(k)
o
kℓcell ≪1
con ains only e en-o de e ms:
ω2(k) = c2k2"1 + X
n≥1
β2n(ˆ
k)(kℓcell)2n#,
(29)
wi hou odd-o de co ec ions like
k3, k5, . . .
. Specically, unde iso opic app oxima ion,
he dominan co ec ion is
ω2=c2k21 + β2(kℓcell)2,
(30)
i.e., he lowes -o de Lo en z iola ion o G a i yQCA dispe sion mus be o
k4
ype, no
odd-o de ypes like
k3
.
7
3.3 Theo em 3.3 (Uppe Bound on QCA La ice Spacing om
Pa ame e ized Dispe sion Cons ain s)
Theo em 3.3
(Uppe Bound on QCA La ice Spacing om Pa ame e ized Dispe sion
Cons ain s)
.
Conside pa ame e ized dispe sion ela ion
ω2=c2k2+αdispk2+n,
(31)
whe e
n≥0
,
αdisp
is a cons an wi h app op ia e dimensions. Assuming
n= 2
ma ches
QCA iso opic dominan co ec ion, i.e.,
ω2=c2k21 + β2(kℓcell)2=c2k2+αdispk4,
(32)
hen
αdisp =c2β2ℓ2
cell.
(33)
I LIGO/Vi go/KAGRA join analysis gi es a some condence le el
|αdisp|≲c2
M2
∗
,
(34)
hen QCA la ice spacing sa ises
ℓcell ≲M−1
∗|β2|−1/2.
(35)
Typically, o ep esen a i e
n= 2
dispe sion cons ain s, li e a u e gi es eec i e
ene gy scale
M∗
a leas in he
1013

1015 GeV
ange, co esponding o
ℓcell ≲10−29

10−31 m (β2∼1),
(36)
indica ing ha i QCA disc e e s uc u e exis s in he uni e se, i s la ice spacing mus
be a leas dozens o o de s o magni ude smalle han cu en ly di ec ly de ec able leng h
scales.
3.4 Theo em 3.4 (G a i yQCA Causal Consis ency and Lo en z
Viola ion Bounds)
Theo em 3.4
(G a i yQCA Causal Consis ency and Lo en z Viola ion Bounds)
.
In he
unied ime scale, bounda y ime geome y, and NullModula double co e amewo k,
assume:
1. Gene alized en opy ex emum and non-nega i e second-o de ela i e en opy on
small causal diamonds hold, equi alen o local Eins ein equa ions and QNEC/QFC in-
equali ies;
2. Bounda y modula ow, sca e ing phase, and geome ic ime align unde unied
scale;
3. Ligh cones o G a i yQCA model and geome ic ligh cones a e consis en o
O(10−15)
in LIGO/Vi go band;
4. NullModula double co e has no
Z2
holonomy anomaly.
Then g a i a ional wa e dispe sion co ec ions mus sa is y:
1. Only e en-o de
(kℓcell)2n
ype e ms a e allowed; non-ze o odd-o de
k2n+1
e ms
would necessa ily in oduce o bidden hal -pe iod phases in NullModula s uc u e, io-
la ing condi ion 4;
8
2. G oup eloci y de ia ion sa ises Planck-scale supp ession law

δ g(ω)
c
≲O(ωℓcell)2;
(37)
3. Rela i e de ia ion o unied ime scale densi y sa ises
|δκGW(ω)|
κ(0)
GW(ω)≲O(ωℓcell)2,
(38)
nume ically no exceeding
O(10−15)
magni ude in cu en obse a ion bands, compa ible
wi h speed and dispe sion cons ain s om GW170817 and GWTC-3.
4 P oo s
This sec ion p o ides p oo s o de i a ion ou lines o Theo ems 3.13.4. De ailed calcu-
la ions and echnical lemmas a e in he Appendix.
4.1 P oo o Theo em 3.1: Dispe sion and Unied Time Scale
Densi y
In 1D simplied case, assume inciden plane wa e p opaga es in a medium o leng h
L
,
wi h dispe sion ela ion
ω(k)
and g oup eloci y
g(ω) = ∂ω/∂k
. Sca e ing ma ix can
be w i en as
S(ω) =  (ω) ′(ω)
(ω) ′(ω), (ω) = | (ω)|expiϕ(ω),
(39)
whe e
ϕ(ω)
is ansmission phase. Wigne g oup delay is
τ(ω) = ∂ωϕ(ω).
(40)
In he weak sca e ing limi whe e eec ion is negligible, ansmission phase app ox-
ima ely equals p opaga ion phase o plane wa e in medium:
ϕ(ω)≈k(ω)L, τ(ω)≈L ∂ωk(ω) = L
g(ω).
(41)
In mul i-channel case, Wigne Smi h ma ix
Q(ω) = −iS†(ω)∂ωS(ω)
(42)
eigen alues gi e g oup delays o each channel, ace is hei sum. I he e a e
N
equi alen
channels, hen
Q(ω)≈NL
g(ω).
(43)
Unied ime scale densi y is dened as
κ(ω) = 1
2π Q(ω)≈N
2π
L
g(ω).
(44)
9
packages (e.g., Bilby), nume ical examples can be ob ained by simple modica ion o
exis ing MDR analysis sc ip s. Symbolic de i a ion and con inuum limi calcula ions can
be ep oduced using gene al algeb a so wa e (Ma hema ica, Py hon/SymPy).
Code A ailabili y
Code a ailabili y s a emen is consis en wi h he ex abo e.
Re e ences
[1] B. P. Abbo e al. (LIGO Scien ic Collabo a ion and Vi go Collabo a ion), G a -
i a ional wa es and gamma- ays om a bina y neu on s a me ge : GW170817 and
GRB 170817A,
As ophys. J. Le .
848, L13 (2017).
[2] T. Bake , E. Bellini, P. G. Fe ei a, M. Lagos, J. Nolle , I. Sawicki, S ong Con-
s ain s on Cosmological G a i y om GW170817 and GRB 170817A,
Phys. Re .
Le .
119, 251301 (2017).
[3] S. Mi sheka i, N. Yunes, C. M. Will, Cons aining Lo en z- iola ing, Modied Dis-
pe sion Rela ions wi h G a i a ional Wa es,
Phys. Re . D
85, 024041 (2012).
[4] N. V. K ishnendu, K. G. A un, C. K. Mish a, e al., Tes ing Gene al Rela i i y
wi h G a i a ional Wa es, e iew epo o LVK es s o GR (2021).
[5] C. Gong, T. Zhu, R. Niu, Q. Wu, J.-L. Cui, X. Zhang, W. Zhao, A. Wang, G a i-
a ional wa e cons ain s on non-bi e ingen dispe sions o g a i a ional wa es due
o Lo en z iola ions wi h GWTC-3,
Phys. Re . D
108, 084024 (2023).
[6] J.-H. Rao, W. Zhao, e al., Simula ion S udy on Cons aining G a i a ional Wa e
P opaga ion Speed and Lo en z Viola ion,
Res. As on. As ophys.
24, 085004
(2024).
[7] X. Liu, V. F. He, T. M. Mikulski, e al., Measu ing he Speed o G a i a ional
Wa es om he Fi s and Second Obse ing Run o Ad anced LIGO and Ad anced
Vi go,
Phys. Re . D
102, 024028 (2020).
[8] M. Sch eck, Lo en z Viola ion in As opa icles and G a i a ional Wa es,
Uni e se
10, 13 (2022).
[9] Q. Wang, W. Zhao, e al., Modied G a i a ional Wa e P opaga ions in Linea ized
G a i y in SME, (2025).
[10] S. Kiyo a, K. Yamamo o, Cons ain on Modied Dispe sion Rela ions o G a i-
a ional Wa es om G a i a ional Che enko Radia ion,
Phys. Re . D
92, 104036
(2015).
[11] C. de Rham, G a i a ional Rainbows: LIGO and Da k Ene gy a i s Cu o,
Phys.
Re . Le .
121, 221101 (2018).
[12] Q. Gao, Y. Gong, e al., Cons ain on he mass o g a i on wi h g a i a ional
wa es,
Sci. China Phys. Mech. As on.
66, 220412 (2023).
16

[13] LIGOVi go Collabo a ion, A new cons ain on he mass o 'g a i on', (2019).
[14] L. Mlodinow, e al., Bounds on Quan um Cellula Au oma on La ice Spacing om
Da a on Lo en z Viola ion, (2025).
[15] G. M. D'A iano, N. Mosco, A. Tosini, Weyl, Di ac and Maxwell Quan um Cellula
Au oma a,
Phys. Re . A
93, 062337 (2016).
[16] T. A. B un, J. Ha ing on, M. M. Wilde, De ec ing Disc e e Space ime ia Ma e
In e e ome y,
Phys. Re . D
99, 015012 (2019).
[17] A. F. Fe a i, M. Gomes, J. R. Nascimen o, e al., Lo en z Viola ion in he Lin-
ea ized G a i y,
Phys. Le . B
652, 174 (2007).
[18] I. Ha y, S. Nissanke, P obing he Speed o G a i y wi h LVK, LISA, and Join
Obse a ions,
Gen. Rela i . G a i .
54, 27 (2022).
[19] N. Lou el, e al., P obing Modied G a i a ional-Wa e Dispe sion wi h Bu s s,
(2025).
[20] M. A ola, e al., G a i a ional and Elec omagne ic Che enko Radia ion wi h
Lo en z-Viola ing Modied Dispe sion Rela ions, (2024).
A F om GR Linea Pe u ba ions o GW Sca e ing
Ma ix
A.1 Linea ized Eins ein Equa ions and Mode Decomposi ion
On backg ound me ic
g(0)
µν
, conside small pe u ba ion
hµν
, in oducing gauge condi ion
∇µhµν = 0, hµµ= 0,
(72)
linea ized Eins ein equa ions a e
□hµν + 2R(0)
µανβhαβ = 0.
(73)
On a backg ound
g(0)
µν =ηµν
,
R(0)
µανβ = 0
, equa ion degene a es o
□hµν = 0,
(74)
plane wa e solu ion is
hµν( , x) = ϵµν(ˆ
k) e−i(ω −k·x), ω2=c2k2.
(75)
In sphe ically symme ic s a ic backg ound (e.g., Schwa zschild ex e io ), p ojec ing
pe u ba ion on o ReggeWheele o Ze illi modes educes o adial equa ion
−d2ψℓ
d 2
∗
+Ve ,ℓ( ∗)ψℓ=ω2ψℓ,
(76)
whe e
∗
is o oise coo dina e,
Ve ,ℓ
is eec i e po en ial. Bounda y condi ions a e
ψℓ( ∗)∼


e−iω ∗+Aou
ℓe+iω ∗, ∗→ −∞,
Bou
ℓe+iω ∗, ∗→+∞.
(77)
A e no maliza ion, sca e ing coecien s
Sℓ(ω)
and co esponding phase shi s
δℓ(ω)
a e ob ained, cons i u ing angula momen um componen s o sca e ing ma ix
SGW(ω)
.
17
A.2 Wigne Smi h Ma ix and G oup Delay
Fo each
ℓ
and pola iza ion, dene channel ampli udes
ain, aou
, such ha
aou (ω) = SGW(ω)ain(ω),
(78)
SGW(ω)∈U(NGW)
. Wigne Smi h ma ix is dened as
QGW(ω) = −iS†
GW(ω)∂ωSGW(ω).
(79)
I
SGW(ω)
can be diagonalized as
SGW(ω) =
NGW
X
j=1
e2iδj(ω)Πj,
(80)
hen
QGW(ω)=2X
j
∂ωδj(ω) Πj, τj(ω)=2∂ωδj(ω).
(81)
In a -eld a backg ound limi , ela ion be ween de i a i es o phase shi s, p opa-
ga ion dis ance
L
, and g oup eloci y
g(ω)
is
τj(ω)≈L
g(ω)+cj(ω),
(82)
whe e
cj(ω)
is equency slowly a ying e m ela ed o local sca e ing. Taking ace
and igno ing
cj(ω)
con ibu ion, we ob ain ela ion be ween
κGW(ω)
and
g(ω)
in main
ex .
B G a i yQCA Con inuum Limi and Dispe sion Ex-
pansion
B.1 One-Dimensional Simplied QCA Model
Conside 1D la ice
Λ = Z
, cellula Hilbe space
Hx=C2
ep esen ing wo pola iza ions,
spin ope a o s deno ed by Pauli ma ices
σi
. Dene wo ypes o local ga es:
1. Hopping ga e
Uhop
, exchanging ampli udes be ween adjacen cells:
Uhop =Y
x
exp−iθ(|x+ 1⟩ ⟨x| ⊗ σz+ h.c.);
(83)
2. Cu a u e ga e
Ug a
, applying local phase on each cell:
Ug a =Y
x
exp−iϕ(ˆp)σz,
(84)
whe e
ϕ(ˆp)
is some unc ion o momen um ope a o .
O e all upda e is
U=Ug a Uhop.
(85)
In momen um ep esen a ion,
U(k)
can be w i en as
U(k) = exp−iHe (k)∆ ,
(86)
18
whe e
He (k) = ckσz+γ2k3ℓ2
cellσz+O(k5ℓ4
cell),
(87)
c
and
γ2
a e cons an s de e mined by
θ, ϕ
.
Spec um o
H2
e
:
ω2=H2
e /∆ 2=c2k2h1+2γ2
ck2ℓ2
cell +O(k4ℓ4
cell)i,
(88)
hus
β2= 2γ2/c,
(89)
ob aining dispe sion coecien exp ession o 1D case in main ex .
B.2 High-Dimensional and Aniso opic Gene aliza ion
In high dimensions,
U(k)
is a mul i a ia e unc ion, i s spec um can be w i en as
ω2(k) = c2k2"1 + X
n≥1
β2n(ˆ
k)(kℓcell)2n#.
(90)
Aniso opy is embodied by angula dependence o
β2n(ˆ
k)
. I la ice and ga e sym-
me y is sucien ly high (e.g., cubic la ice and iso opic local ga es), hen in low-o de
app oxima ion
β2n(ˆ
k)≈β2n
can be ea ed as cons an . Fo g a i a ional wa e obse -
a ions, angula aniso opy can be eec i ely smoo hed ou by a e aging o e mul iple
e en s and di ec ions; i s esidual eec s can be used as ad anced me ics o es ne
QCA s uc u es.
C Nume ical Illus a ion o Dispe sion Pa ame e s, Ob-
se a ional Cons ain s, and QCA La ice Spacing
C.1
n= 2
Type Dispe sion and Ene gy Scale
Conside
ω2=c2k2+αdispk4, αdisp =σc2
M2
∗
,
(91)
whe e
M∗
is ene gy scale. Using na u al uni s
c=ℏ= 1
, con e sion is
1 GeV−1≈
2×10−16 m
.
I obse a ional cons ain gi es
M∗≳1014 GeV,
(92)
hen
M−1
∗≲10−14 GeV−1≈2×10−30 m.
(93)
In QCA mapping,
ℓcell ≲M−1
∗|β2|−1/2,
(94)
i
β2∼1
, hen
ℓcell ≲2×10−30 m,
(95)
abou
105
imes Planck leng h
ℓPl ∼10−35 m
.
I u u e obse a ions aise
M∗
o
1015

1016 GeV
, hen
ℓcell
uppe bound will u he
d op o
10−31

10−32 m
ange.
19
C.2 Compa ison wi h GW170817 Speed Cons ain
GW170817 and GRB 170817A gi e

g
c−1≲10−15,
(96)
unde condi ion
∼102Hz
,
L∼40 Mpc
, co esponding o

δ g
c
≲10−15.
(97)
In QCA model,
δ g
c≃β2(kℓcell)2, k ∼2π
c∼10−6m−1,
(98)
so
ℓcell ≲10−7.5
p|β2|m∼10−8m (β2∼1),
(99)
his is an ex emely loose uppe bound. Wha uly d i es
ℓcell
in o
10−29

10−31 m
ange
is cumula i e dispe sion analysis o wa e o m phase, no simple a i al ime die ence
measu emen . This explains why GWTC-3 le el mul i-e en s a is ical analysis is needed
o ob ain s ong cons ain s on high-dimensional ope a o s and QCA la ice spacing.
C.3 Comp ehensi e Cons ain s wi h EM and Ma e Expe i-
men s
Elec omagne ic and ma e expe imen s es Lo en z iola ion a highe ene gies and
longe baselines, p o iding cons ain s on
αdisp
o SME coecien s ha can each ex eme
p ecision. Con e ing hese esul s o uppe bounds on QCA la ice spacing, ob ained
ℓcell
uppe bounds o en o e lap wi h g a i a ional wa e cons ain s in
10−29

10−32 m
ange.
This indica es:
1. I Unied Ma ixQCA uni e se model is co ec , uni e se disc e e la ice spacing
is likely loca ed in his in e al o below;
2. G a i a ional wa e channel and elec omagne ic/ma e channels p o ide comple-
men a y and co obo a i e cons ain s, building a unied amewo k o obse a ional
es ing o uni e se disc e e s uc u e.
20