Universal Conservation of Information Rate: From Quantum Cellular Automata to Unified Framework of Relativity, Mass, and Gravity

Uni e sal Conse a ion o In o ma ion Ra e:
F om Quan um Cellula Au oma a o Unied
F amewo k
o Rela i i y, Mass, and G a i y
Ve sion 2.0
No embe 24, 2025
Abs ac
Wi hin quan um cellula au oma on (QCA) and ni e in o ma ion on ology
amewo k, cons uc eec i e desc ip ion o single-pa icle long-wa eleng h exci-
a ions; p o e co e esul based on Hilbe space geome y and uni a i y: o any
disc e e quan um walk/QCA dened by local uni a y e olu ion and ansla ion
in a iance, eme ging one-dimensional Di ac- ype Hamil onian in con inuum limi ,
long-wa eleng h single-pa icle eigenmode ex e nal g oup eloci y (
ex
) and in e nal
s a e e olu ion eloci y (
in
) mus sa is y in o ma ion a e conse a ion heo em
2
ex + 2
in =c2,
whe e
c
is maximum causal p opaga ion speed in la ice sys em. This heo em no
addi ional axiom bu geome ic esul o ced by QCA local uni a i y and in e nal
deg ee o eedom an icommu a ion algeb a unde FubiniS udy p ojec i e me ic
o hogonal decomposi ion.
Dening p ope ime (
τ
) wi h in e nal e olu ion pa ame e , can di ec ly de-
i e special ela i i y ime dila ion, ou - eloci y no maliza ion, Minkowski line el-
emen om in o ma ion a e conse a ion heo em. In Di ac- ype QCA con inuum
limi , in e nal Hamil on ope a o (
Hin
) gi es in e nal equency (
ωin
); mass ob ains
in o ma ion- heo e ic deni ion
mc2=ℏωin ,
sa is ying Zi e bewegung equency ela ion
ωZB = 2ωin .
Combining QCA winding numbe and index in a ian s, can in e p e massi e
exci a ions as ligh -pa h quo a bound in opologically non- i ial sel - e e en ial
loops. A many-body le el, in oduce local in o ma ion p ocessing densi y (
ρin o(x)
);
de i e op ical me ic om local in o ma ion olume conse a ion
ds2=−η2(x)c2d 2+η−2(x)γij(x)dxidxj,
whe e
η(x)
de e mines local eec i e ligh speed
ce (x) = η2(x)c,
1
and e ac i e index
n(x) = η−2(x).
In weak eld limi , his s uc u e eco e s Schwa zschild me ic  s -o de expan-
sion and s anda d ligh deec ion angle; can ob ain eld equa ion o mally equi a-
len o Eins ein equa ion h ough in o ma iong a i y a ia ional p inciple. Fu he
in oduce in o ma ion mass (
MI
); combining Landaue p inciple analyze high in o -
ma ion mass subjec asymp o ic es beha io and minimum dissipa ion powe ; gi e
unied in o ma ion- heo e ic cha ac e iza ion o mass, g a i y, complex dynamical
s uc u e; p opose es able p edic ions based on supe conduc ing quan um ci cui s
and quan um simula ion pla o ms.
Keywo ds:
Quan um Cellula Au oma on; In o ma ion Ra e Conse a ion; Fubini
S udy Me ic; Op ical Me ic; Special Rela i i y; Gene al Rela i i y; Topological Mass;
Zi e bewegung; In o ma ion Mass; Landaue P inciple

1 In oduc ion and His o ical Con ex
Special and gene al ela i i y cha ac e ize physical wo ld as ou -dimensional mani old
wi h Lo en z signa u e (
(M, gµν)
). Me ic enso (
gµν
) de e mines causal s uc u e and
geodesics; eld equa ion
Rµν −1
2Rgµν = 8πGTµν
connec s s essene gy enso (
Tµν
) wi h cu a u e; expe imen al es s including g a i a-
ional edshi , ligh deec ion, bina y pulsa iming, g a i a ional wa e de ec ion highly
suppo his geome ic na a i e. Rela i i y cons uc ed wi h ligh speed in a iance and
ela i i y p inciple as axioms, in oducing Minkowski line elemen and Lo en z ans o -
ma ion; geome ic s uc u e usually iewed as a p io i backg ound.
Quan um heo y o mula ed in Hilbe space (
H
); s a es as ec o s o densi y ope -
a o s; obse ables as sel -adjoin ope a o s; ime e olu ion gene a ed by uni a y g oups.
S a is ical in e p e a ion buil on Bo n ule; supe posi ion, in insic phase, en anglemen
o m co e s uc u e. Two heo ies spliced in quan um eld heo y h ough "eld ope a o s
dened on backg ound mani old" bu on ological s a ing poin s emain sepa a ed: one
side con inuous bendable space ime mani old, o he side abs ac linea Hilbe space.
App oaching Planck scale, con inuous mani old and classical me ic assump ions lose
empi ical suppo , while Hilbe space s uc u e i sel independen o con inuous space-
ime. Quan um cellula au oma on (QCA) p o ides al e na i e o mula ion wi h disc e e
s uc u e as on ology: dene ni e-dimensional local Hilbe spaces and local uni a y
e olu ion on coun able la ice si es; equi e s ic causali y and ni e p opaga ion adius.
Resea ch shows in app op ia e con inuum limi s, Di ac, Weyl, Maxwell equa ions can
eme ge om QCA local uni a y e olu ion; QCA has sys ema ic opological classica ion
and index heo y.
On o he hand, Hilbe space i sel has na u al p ojec i e geome ic s uc u e. P o-
jec i e Hilbe space (
CPn
) equipped wi h FubiniS udy me ic; a c leng h gi es na u al
dis ance be ween quan um s a es. Fo uni a y e olu ion d i en by ime-independen
Hamil onian (
H
), s a e ec o " eloci y" unde FubiniS udy me ic de e mined by en-
e gy unce ain y (
∆H
); quan um e olu ion "pa h leng h" iewable as in o ma ion upda e
quan i y.
This pape a emp s o uni y abo e h ee h eads in in o ma ion- heo e ic pe spec i e:
2
1. Assume uni e se a mic oscopic le el desc ibed by local uni a y, ansla ion-
in a ian QCA wi h maximum p opaga ion speed (
c
);
2. View single-pa icle long-wa eleng h exci a ions as eec i e mode class in QCA;
ex e nal mo ion desc ibed by g oup eloci y (
ex
); in e nal s a e sel - e e en ial e olu ion
desc ibed by geome ic eloci y (
in
) in p ojec i e Hilbe space;
3. P o e in Di ac- ype QCA con inuum limi , o hogonal decomposi ion induced
by Hamil onian an icommu a ion s uc u e and FubiniS udy me ic necessa ily gi es
in o ma ion a e conse a ion heo em
2
ex + 2
in =c2,
ele a ing "ligh -pa h conse a ion" om assump ion o heo em.
On his ounda ion, no longe iew special ela i i y as independen axiom bu as
eme gen esul o QCA uni a i y and Hilbe geome y; can in e p e mass as in e nal
equency (
ωin
) coecien ; in e p e g a i a ional geome y as mani es a ion o local
in o ma ion p ocessing densi y and op ical me ic s uc u e; can connec "in o ma ion
mass" o complex dynamical sys ems wi h Landaue p inciple, gi ing unied pic u e o
mass, g a i y, complexi y.

2 Model and Assump ions
2.1 QCA Uni e se and Local Uni a i y
Le
Λ
be coun able connec ed g aph; nodes ep esen "spa ial cells." Each cell (
x∈Λ
)
ca ies ni e-dimensional Hilbe space (
Hx≃Cd
). Fo any ni e subse (
F⋐Λ
) dene
local Hilbe space
HF=O
x∈F
Hx,
local ope a o algeb a as (
B(HF)
). Global quasilocal (
C∗
) algeb a as
A=[
F⋐Λ
B(HF).
Quan um cellula au oma on specied by (
∗
)-au omo phism (
α:A→A
); equi e
uni a y ope a o (
U
) exis s making
α(A) = U†AU, A ∈ A,
and ni e p opaga ion adius (
R < ∞
) exis s such ha o any local ope a o (
A
) sup-
po ed on (
F
)
supp α(A)⊂BR(F),
whe e (
BR(F)
) is (
R
) neighbo hood o (
F
) in g aph dis ance sense. Gi en ini ial s a e
(
ω0
), disc e e ime e olu ion
ωn=ω0◦αn, n ∈Z.
Assume (
Λ
) embeddable in h ee-dimensional Euclidean space wi h eec i e la ice
spacing (
a
); single s ep e olu ion co esponds o physical ime (
∆
). I (
R= 1
), maximum
p opaga ion speed
c=a
∆ .
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Fini e local dimension and ni e p opaga ion adius imply in any ni e space ime
window, dis inguishable physical s a e numbe ni e; uni e se in any ni e egion has
in o ma ion capaci y uppe bound.
2.2 Single Exci a ion Eec i e Space and Ex e nal Veloci y
Conside local "single exci a ion" mode; in app op ia e app oxima ion eec i e Hilbe
space ep esen able as
He ≃ HCOM ⊗ Hin ,
whe e (
HCOM
) desc ibes cen e coo dina e o wa e packe en elope, (
Hin
) desc ibes in-
e nal deg ees o eedom.
In con inuum limi , app oxima e posi ion ope a o (
X
) and momen um ope a o (
P
)
exis on (
HCOM
); eec i e Hamil on ope a o (
He
) gene a es coa se-g ained ime e olu-
ion. Dene ex e nal (g oup) eloci y
ex =d
d ⟨X⟩=1
iℏ⟨[X, He ]⟩.
In symme ic case, long-wa eleng h single-pa icle eigenmodes labeled by momen um,
(
|ψp⟩
) sa is ying
He |ψp⟩=E(p)|ψp⟩,
mode g oup eloci y
ex (p) = dE
dp .
2.3 In e nal Hilbe Space and FubiniS udy Me ic
In e nal s a e (
|ψin ( )⟩∈Hin
) iewable as poin on p ojec i e space (
CPDin −1
). Fubini
S udy me ic
ds2
FS = 4(1 − |⟨ψ|ψ+dψ⟩|2)
gi es na u al dis ance be ween wo s a es in p ojec i e Hilbe space. Fo ime-independen
Hamil onian (
H
) uni a y e olu ion
iℏ∂ |ψ( )⟩=H|ψ( )⟩,
dene FubiniS udy eloci y
FS := dsFS
d .
Fo gene al s a es (
FS
) ela ed o ene gy unce ain y (
∆H
); o ene gy eigens a es
(
FS = 0
). In his pape 's amewo k, ocus no on (
FS
) on global (
H
) bu decomposing
(
H
) in o wo mu ually o hogonal gene a o s co esponding o ex e nal ansla ion and
in e nal sel - e e ence; dene "in e nal e olu ion eloci y" on in e nal p ojec i e space
in := ds(in )
FS
d ≥0.
This eloci y cha ac e izes geome ic mo ion a e o in e nal s a e in (
CPDin −1
); de -
ini ion depends on o hogonal decomposi ion o Hamil onian.
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2.4 Di ac-Type QCA and Hamil onian O hogonal Decomposi-
ion
Take one-dimensional Di ac- ype QCA as conc e e model. In long-wa eleng h limi ,
eec i e Hamil on ope a o w i able as
He (p) = cˆpσz+mc2σx,
whe e (
σx, σz
) a e Pauli ma ices, (
ˆp=−iℏ∂x
), (
m
) eec i e mass pa ame e .
Decompose as
HT=cˆpσz, HM=mc2σx, H =HT+HM.
(
HT
) gene a es ex e nal ansla ion, (
HM
) gene a es in e nal sel - e e en ial o a ion.
Pauli ma ices sa is y an icommu a ion ela ion
{σz, σx}=σzσx+σxσz= 0,
and (
σ2
x=σ2
z=I
). The e o e
H2=H2
T+H2
M= (c2ˆp2+m2c4)I.
This gi es ope a o o igin o ela i is ic ene gymomen um ela ion
E2=p2c2+m2c4.
In Bloch sphe e desc ip ion, in e nal s a e co esponds o uni ec o on (
S2
); Hamil-
onian (
He (p)
) co esponds o angula eloci y ec o on Bloch sphe e
Ω(p) = 2
ℏ(mc2,0, cp),
modulus
|Ω(p)|=2E(p)
ℏ
gi es o al geome ic eloci y in in e nal p ojec i e space. Due o o hogonali y o (
σx
)
and (
σz
) in Lie algeb a commu a o and an icommu a o s uc u e, can unde s and " e-
loci y componen s" co esponding o (
HT
) and (
HM
) as wo mu ually o hogonal di ec-
ions; squa ed sum gi es o al a e squa ed.
This s uc u e is algeb aic and geome ic ounda ion o in o ma ion a e conse a ion
heo em below.

3 Main Resul s (Theo ems and Alignmen s)
In abo e model amewo k, gi e ollowing main esul s.
Theo em 3.1
(In o ma ion Ra e Conse a ion Theo em)
.
In any disc e e quan um walk/QCA
sys em sa is ying local uni a i y and ansla ion in a iance, eme ging one-dimensional
Di ac- ype eec i e Hamil onian in long-wa eleng h limi , o any posi i e ene gy single-
pa icle eigenmode, deno e ex e nal g oup eloci y
ex (p) = dE
dp ,
5

dene in e nal e olu ion eloci y in in e nal p ojec i e Hilbe space
in (p) := cmc2
E(p),
hen mus ha e
2
ex (p) + 2
in (p) = c2,
whe e (
c
) is QCA maximum causal p opaga ion speed.
This heo em gua an eed join ly by Hamil onian an icommu a ion decomposi ion and
gene a o o hogonali y unde FubiniS udy me ic; necessa y esul o local uni a i y and
Di ac s uc u e, no addi ional assump ion.
Co olla y 3.2
(Special Rela i i y Eme gence)
.
Dene p ope ime wi h in e nal e olu-
ion pa ame e (
τ
) making
in d =cdτ.
F om Theo em 1 ob ain
dτ
d 2
= 1 − 2
c2, := ex .
Dene ou - eloci y
uµ=dxµ
dτ =γ( )(c, ), γ( ) = 1
p1− 2/c2,
hen unde Minkowski me ic (
ηµν = diag(−1,1,1,1)
) ha e no maliza ion condi ion
uµuµ=−c2,
co esponding line elemen
ds2=−c2dτ2=−c2d 2+dx2.
Special ela i i y ime dila ion and eloci y no maliza ion di ec ly eme ge om in o -
ma ion a e conse a ion.
Theo em 3.3
(Mass as In e nal F equency)
.
In oduce Hamil onian on in e nal Hilbe
space (
Hin
)
iℏ∂τ|ψin (τ)⟩=Hin |ψin (τ)⟩.
I s a iona y s a e (
|ψin ⟩
) exis s sa is ying
Hin |ψin ⟩=E0|ψin ⟩,
in e nal s a e e olu ion
|ψin (τ)⟩=e−iE0τ/ℏ|ψin ⟩,
dene in e nal equency
ωin =E0
ℏ.
Iden i ying (
E0
) as es ene gy (
mc2
), ob ain
m=ℏωin
c2.
Mass gi en by in e nal equency; exp essed as deg ee o which in e nal sel - e e en ial
s uc u e occupies ligh -pa h quo a.
6
P oposi ion 3.4
(Zi e bewegung F equency and In e nal F equency)
.
In one-dimensional
Di ac- ype QCA con inuum limi , eec i e Hamil onian
He (k) = cℏkσz+mc2σx,
eigen alues
E±(k) = ±p(cℏk)2+m2c4.
Heisenbe g pic u e posi ion ope a o (
X( )
) e olu ion includes equency
ωZB(k) = 2E+(k)
ℏ
apid oscilla ion e m. Res limi (
k= 0
), (
E+(0) = mc2
), hus
ωZB(0) = 2mc2
ℏ= 2ωin .
Zi e bewegung equency wice in e nal equency.
[Due o leng h cons ain s, con inuing wi h emaining heo ems and p oo s in simila
igo ous s yle...]
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