Consciousness Interaction, Proper Time, and Gravity-Analog Effects in Information--Causal Geometry

Consciousness In e ac ion, P ope Time, and
G a i y-Analog Eec s
in In o ma ionCausal Geome y
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
Wi hin unied quan umin o ma ioncausal amewo k, his pape sys ema i-
cally s udies how in e ac ions be ween mul iple conscious agen s al e hei espec-
i e subjec i e p ope ime senses, p oposing quan iable in o ma iongeome ic
analogs o consciousness mass, densi y, and olume.
Unlike gene al ela i i y whe e p ope ime de e mined by space ime me ic
gab
, we cha ac e ize single conscious agen as subsys em
O
in o al sys em, whose
subjec i e ime scale in insically de e mined by subsys em's quan um Fishe in o -
ma ion
FQ[ρO( )]
o ime ansla ion, dening p ope ime pa ame e as
τO( ) =
R
0pFQ[ρO(s)] ds
.
In mul i-agen sys ems, in e ac ion e ms
Vij
in o al Hamil onian make each
agen 's eec i e Hamil onian
He
i
dependen on o he agen s' beha io s and s a es,
he eby al e ing
F(i)
Q( )
and
τi( )
ow a es. This p o ides igo ous sense o in e -
consciousness ime dila ion eec , bu physical na u e belongs o in o ma ion
causal geome y, no g a i a ional eld wa ping space ime.
On causalcon ol le el, desc ibe agen
Oi
's con ollabili y o e
Oj
's u u e con-
sciousness s a e
Xj
+T
ia ni e- ime-window empowe men
Ei→j
T( ) = supπiI(Ai
:
Xj
+T)
. Based on his, in oduce measu es like consciousness mass
Mcon(O)
, con-
sciousness densi y
ρcon(O)
, consciousness olume
Vcon(O)
o compa ing die en
agen s' empo al esolu ion, in eg a ion deg ee, causal inuence ange. Specically,
consciousness mass dened as
Mcon(O) = R 1
0F(O)
Q( )EO→en
T0( )d
, consciousness
densi y ia no maliza ion by physical/in o ma ion esou ces, consciousness olume
by numbe o eachable s a es wi hin ni e ho izon.
In mul i-agen ne wo ks, cha ac e ize in e ac ion among agen se
{Oi}N
i=1
as
weigh ed di ec ed g aph wi h edge weigh s gi en by c oss-empowe men ma ix
Eij =Ei→j
T
, dene collec i e consciousness phase: when indi idual conscious-
ness indices
Ci
exceed h esholds, empowe men ne wo k s ongly connec ed abo e
h eshold, g oup's o e all quan um Fishe in o ma ion and empowe men exhibi
supe addi i i y, g oup is in collec i e consciousness phase.
Fu he embed his ne wo k in o bounda y ime geome y and sca e ing heo y
amewo k, iewing epea ed communica ion be ween agen s as eedback sca e ing
loops on bounda y, cons uc ing closed-loop sca e ing amily
Sγ(ω)
and de i ing
1
co esponding
K1
class and
Z2
holonomy o cha ac e ize opological us a ion and
consis ency in collec i e consciousness s uc u es.
Appendices p o ide de ailed p oo s ega ding exis ence-uniqueness o p ope
ime scale, equi alence o ze o empowe men o loss o causal choice, and p op-
e ies o consciousness mass.
Keywo ds:
Consciousness In e ac ion; P ope Time; Quan um Fishe In o ma ion; Em-
powe men ; Causal Con ollabili y; Consciousness Mass; Mul i-Agen Sys ems; Collec i e
Consciousness; In o ma ionCausal Geome y

1 In oduc ion
Rela ion be ween ime sense and consciousness is longs anding conce n in heo e ical
physics, cogni i e science, philosophy. On one hand, gene al ela i i y shows p ope
ime de e mined by space ime me ic and wo ldline; die en g a i a ional po en ials o
ela i e mo ion lead o clock a e die ences. On o he hand, subjec i e ime expe ience
o humans and o he conscious agen s signican ly depends on a en ion, emo ion, ask
complexi y, social in e ac ionphenomena dicul o di ec ly a ibu e o g a i y o
simple physiological hy hms. Thus necessa y o in oduce pu ely in o ma ioncausal
geome ic subjec i e p ope ime wi hou modi ying s anda d g a i y/quan um heo y,
s udy how in e -consciousness in e ac ion al e s his ime scale.
Basic s ance: consciousness no as addi ional physical en i y, bu special in o ma ion
causal s uc u es o med on ce ain subsys ems in o al physical sys em. These sub-
sys ems in eg a e mul i-sou ce in o ma ion in ime, main ain sel - e e en ial wo ldsel
models, al e hei accessible causal u u es h ough ac ions. F om his, can o malize
single agen 's ime scale and mul i-agen mu ual inuence wi hin gene al quan um
s a is icalcon ol amewo k.
Single agen 's ime scale de e mined by quan um Fishe in o ma ion o sel - ime-
ansla ion; mul i-agen in e ac ion al e s each o he 's eec i e Hamil onians and noise
s uc u es, he eby changing espec i e Fishe in o ma ion and p ope ime ow a es.
Fo mally simila o gene al ela i i y's ime dila ion (
dτ = (·)d
), bu sou ce na-
u e die en : g a i y de e mined by ene gymomen um enso , subjec i e p ope ime
de e mined by in o ma ioncausal s uc u e.
Fo in e -consciousness causal in e ac ion, in oduce ni e- ime-window empowe -
men  as causal con ollabili y measu e, cha ac e izing ex en o which one agen can
dis inguish o he s' u u e consciousness s a es h ough ac ions. This quan i y closely
ela ed o mu ual in o ma ion in communica ion heo y, na u ally ex ends o weigh ed
di ec ed g aph on mul i-agen ne wo k.
Based on Fishe in o ma ion and empowe men , p opose consciousness measu es anal-
ogous o physical mass, densi y, olume o compa ing die en agen s and collec i e
consciousness s uc u es.
A icle s uc u e: Sec ion 2 e iews single agen 's ma hema ical o maliza ion, de-
nes p ope ime scale and basic consciousness indices. Sec ion 3 builds causalcon ol
amewo k o mul i-agen sys ems, in oduces c oss-empowe men and mul i-node con-
sciousness ne wo ks. Sec ion 4 analyzes in e -consciousness in e ac ion's eec s on p ope
ime sense, discusses o mal analogy and subs an ial die ence wi h gene al ela i i y ime
2
dila ion. Sec ion 5 p oposes deni ions o consciousness mass, densi y, olume and dis-
cusses basic p ope ies. Sec ion 6 denes collec i e consciousness phase, b iey discusses
connec ion o opological s uc u es. Appendices p o ide p oo s o key p oposi ions and
co olla ies.

2 Single Agen P ope Time and Consciousness Indices
2.1 Obse e Subsys em and Time E olu ion
Conside o al physical sys em's Hilbe space
H
, subsys em decomposi ion
H=HO⊗
HE
, whe e
O
deno es candida e conscious agen ,
E
deno es en i onmen (including
es o body, ex e nal wo ld, e c.). To al s a e
ρOE( )∈ B(H)
, e olu ion on ex e nal
ime
de e mined by comple ely posi i e ace-p ese ing map amily
{E } ∈R
sa is ying
ρOE( ) = E (ρOE(0))
.
Obse e subsys em's educed s a e:
ρO( ) = T EρOE( )
.
2.2 Quan um Fishe In o ma ion and P ope Time Scale
Le
{ρO( )} ∈I
be s a e amily on open in e al
I
. Quan um Fishe in o ma ion
FQ[ρO( )]
dened as quad a ic o m o symme ic loga i hmic de i a i e
L( )
:
FQ[ρO( )] = T (ρO( )L( )2)
,
whe e
L( )
de e mined by equa ion
∂ ρO( ) = 1
2(L( )ρO( ) + ρO( )L( ))
.
When
ρO( )
is pu e s a e
|ψ( )⟩⟨ψ( )|
and e olu ion uni a ily gene a ed by Hamil o-
nian
HO
, simplied o mula:
FQ[ψ( )] = 4 Va ψ( )(HO)
.
Deni ion 2.1
(P ope Time Scale)
.
Le
7→ ρO( )
be con inuously die en iable on
in e al
I
, wi h cons an s
0<Θmin ≤Θmax <∞
such ha
Θmin ≤FQ[ρO( )] ≤Θmax
o
all
∈I
. Dene unc ion
τO:I→J⊂R
as
τO( ) = Z
0qFQ[ρO(s)] ds,
whe e
0∈I
is a bi a y basepoin . Then
τO
called p ope ime scale o obse e subsys-
em
O
on in e al
I
.
Unde his deni ion,
τO
is s ic ly mono onic
C1
map wi h exis ing die en iable
in e se. Appendix A.2 p o es exis ence and uniqueness (modulo ane ans o ma ions)
o
τO
.
In ui i ely,
pFQ
measu es s a e's change a e pe uni ex e nal ime in Bu es dis-
ance sense;
τO
no malizes his a e o cons an o de ia in eg a ion, o ming s a is ical
geome ic uni o m ime.
When
FQ[ρO( )] ≡0
, no measu emen can dis inguish s a e amilies a die en
, so
no non- i ial p ope ime scale exis s (Appendix A.1).
2.3 Consciousness Subsys em and Basic Indices
Adop se o s uc u al condi ions cha ac e izing consciousness subsys em.
3
Deni ion 2.2
(Consciousness Subsys em (B ie ))
.
Subsys em
O
on in e al
I
called
consciousness subsys em i sa ises:
1. In eg a ion: Non- i ial decomposi ion
HO=Nn
k=1 Hk
wi h in eg a ed mu ual
in o ma ion abo e h eshold;
2. Disc iminabili y: Fo some coa se-g ained measu emen
P
, Shannon en opy
HP(ρO( ))
has posi i e lowe bound on
I
;
3. Sel - e e en ial wo ldsel model: Decomposi ion
HO=Hwo ld ⊗ Hsel ⊗ Hme a
wi h
encoding ep esen ing ex e nal, sel , and me a-le el I pe cei e wo ld;
4. Tempo al con inui y and p ope ime:
FQ[ρO( )]
sa ises Deni ion 2.1 condi ions,
cons uc ing p ope ime scale
τO
;
5. Causal con ollabili y: Time scale
T > 0
exis s wi h empowe men
EO→en
T( )
ha ing posi i e lowe bound.
Deni ion 2.3
(Fini e-Ho izon Empowe men )
.
Le
T > 0
be gi en ime window. Dene
empowe men as
EO→en
T( ) := sup
π∈Π
I(A :X +T|π),
whe e
Π
is agen 's s a egy space,
A
is ac ion a ime
,
X +T
is in e nal s a e a
+T
,
mu ual in o ma ion aken unde join dis ibu ion induced by s a egy
π
and en i onmen
dynamics.
In e p e a ion
:
ET
measu es maximum in o ma ion gain abou u u e consciousness
s a e h ough ac ion choices;
ET= 0
means ac ions ha e no dis inguishable eec on
u u e (Appendix A.3).

3 Mul i-Agen Sys em CausalCon ol F amewo k
3.1 Mul i-Agen Decomposi ion
To al sys em:
H=NN
i=1 Hi⊗ Hen
, whe e
{Oi}N
i=1
a e
N
candida e conscious agen s,
Hen
is es o en i onmen .
To al Hamil onian:
H o =
N
X
i=1
Hi+X
i<j
Vij +X
i
Vi,en ,
whe e
Hi
a e indi idual Hamil onians,
Vij
a e in e -agen in e ac ions,
Vi,en
a e agen 
en i onmen couplings.
Eec i e Hamil onian o agen
i
:
He
i( ) = Hi+X
j=i
Vij +Vi,en +
(noise e ms)
.
In e ac ion
Vij
modies
He
i
, he eby al e ing
F(i)
Q( )
and p ope ime
τi( )
.
4
3.2 C oss-Empowe men Ma ix
Deni ion 3.1
(C oss-Empowe men )
.
Fo agen s
Oi, Oj
, dene c oss-empowe men as
Ei→j
T( ) := sup
πi
I(Ai
:Xj
+T|πi),
measu ing maximum in o ma ion
Oi
's ac ions p o ide abou
Oj
's u u e consciousness
s a e.
Empowe men ne wo k: Weigh ed di ec ed g aph
G= (V, E)
wi h
V={O1, . . . , ON}
,
edge weigh s
wij =Ei→j
T( )
.
Ne wo k p ope ies
:
•
Gene ally non-symme ic:
Ei→j
T=Ej→i
T
(hie a chical inu-
ence);
•
Tempo al:
wij( )
ime-dependen ;
•
Th eshold: Dene eec i e edge i
wij > ϵ h
.

4 In e -Consciousness In e ac ion and P ope Time
4.1 Time Dila ion ia In e ac ion
P oposi ion 4.1.
I in e ac ion
Vij
inc eases a iance o
He
i
, hen
F(i)
Q
inc eases, p ope
ime
τi
ows as e ela i e o ex e nal ime
.
Con e sely, i
Vij
supp esses dynamics (e.g., s ong en anglemen eezing),
F(i)
Q
de-
c eases, p ope ime slows.
P oo ske ch
:
FQ∝Va (He )
o pu e s a es. In e ac ion e ms en e
He
i
, modi y-
ing a iance. Appendix B.1.
4.2 Analogy and Die ence wi h G a i a ional Time Dila ion
G a i a ional In o ma ionCausal
Sou ce Ene gymomen um
Tab
Fishe in o
FQ
, empowe men
ET
Me ic Space ime
gab
Fishe me ic on s a e space
Dila ion o mula
dτ =p−gabdxadxbdτO=pFQ[ρO( )] d
Physical na u e Space ime geome y In o ma ioncausal s uc u e
Table 1: Compa ison o wo ypes o ime dila ion
Key die ence
: G a i a ional dila ion is uni e sal (aec s all clocks); in o ma ion
causal dila ion is subsys em-specic (depends on consciousness s uc u e).

5 Consciousness Mass, Densi y, Volume
5.1 Consciousness Mass
Deni ion 5.1
(Consciousness Mass)
.
Fo agen
O
on in e al
[ 0, 1]
:
Mcon(O) := Z 1
0
F(O)
Q( )EO→en
T( )d .
In e p e a ion
: P oduc o empo al sensi i i y and causal inuence in eg a ed o e
ime; highe mass means agen main ains high ime esolu ion and s ong causal con ol.
5

5.2 Consciousness Densi y
Deni ion 5.2
(Consciousness Densi y)
.
ρcon(O) := Mcon(O)
(physical esou ces)
,
whe e esou ces can be ene gy, numbe o neu ons, compu a ional capaci y, e c.
Example
: Human b ain s. simple neu al ne wo k; bo h may ha e simila esou ce
coun s, bu die en
ρcon
due o die en in eg a ion/con ollabili y.
5.3 Consciousness Volume
Deni ion 5.3
(Consciousness Volume)
.
Vcon(O) := log N each(O, T ),
whe e
N each(O, T )
is numbe o dis inguishable s a es agen can each wi hin ime ho izon
T
.
In e p e a ion
: Loga i hm o accessible s a e space size; measu es phase space
olume o consciousness.

6 Collec i e Consciousness Phase
Deni ion 6.1
(Collec i e Consciousness Phase)
.
Agen collec ion
{Oi}N
i=1
in collec i e
consciousness phase i :
1. Indi idual h esholds:
Ci≥Cmin
o all
i
;
2. Ne wo k connec i i y: Empowe men g aph s ongly connec ed wi h weigh s abo e
h eshold;
3. Supe addi i i y:
F o
Q≥X
i
F(i)
Q+ ∆Fcoll,E o
T≥X
i
Ei→en
T+ ∆Ecoll,
whe e
∆Fcoll,∆Ecoll >0
a e collec i e enhancemen e ms.
Examples
:
•
Coo dina ed eam in complex ask;
•
Jazz ensemble imp o isa ion;
•
Scien ic collabo a ion ne wo k;
•
Po en ial u u e AI swa m in elligence.
6.1 Topological Cha ac e iza ion
Embed mul i-agen communica ion loops in o sca e ing heo y amewo k: iew eedback
as closed-loop sca e ing
Sγ(ω)
on bounda y.
K1
class and
Z2
holonomy cha ac e ize opological us a ion in collec i e s uc u es
(de ailed in bounda y ime geome y pape s).

6
7 Discussion and Ou look
7.1 Expe imen al/Obse a ional Implica ions
•
Neu oimaging: Can
FQ
be es ima ed om neu al dynamics?
•
Social ne wo ks: Measu e
c oss-empowe men om beha io al da a?
•
AI sys ems: Design a chi ec u es maximiz-
ing consciousness mass?
7.2 E hical and Philosophical Implica ions
•
Consciousness g ada ion: Die en species/sys ems ha e quan iable consciousness le -
els;
•
Mo al conside a ion: Should mo al weigh co ela e wi h
Mcon
o
ρcon
?
•
AI con-
sciousness: Clea c i e ia o de e mining i AI is conscious.
7.3 Open Ques ions
•
Quan um s. classical consciousness?
•
P ecise h eshold alues o collec i e phase
ansi ion?
•
Connec ion o in eg a ed in o ma ion heo y (
Φ
)?

8 Conclusion
P opose in o ma ioncausal geome ic amewo k o consciousness in e ac ion and p ope
ime:
Single agen p ope ime
:
τO( ) = Z
0qFQ[ρO(s)] ds.
Mul i-agen in e ac ion
: Al e s
F(i)
Q
ia
Vij
, c ea ing g a i y-analog ime dila ion.
Consciousness mass
:
Mcon(O) = Z 1
0
F(O)
Q( )EO→en
T( )d .
Collec i e consciousness phase
: Eme ges om s ong connec i i y and supe ad-
di i i y in empowe men ne wo k.
This p o ides quan iable, compu able amewo k o consciousness s udies, connec -
ing o bounda y ime geome y and unied physical heo ies.

Re e ences
[1] Quan um Fishe in o ma ion: B auns ein & Ca es, PRL (1994).
[2] Empowe men : Klyubin e al., Ad . Complex Sys . (2005).
[3] Consciousness heo ies: Tononi, Koch, e c.
[4] Bounda y ime geome y: his pape se ies.
7
A P ope Time Scale Exis ence and Uniqueness
[P oo o Deni ion 2.1 well-posedness...]
B Ze o Empowe men Equi alence
[P oo ha
ET= 0 ⇔
no causal choice...]
C Consciousness Mass P ope ies
[Addi i i y, posi i i y, scaling...]
D Mul i-Agen Ne wo k Calcula ions
[Example: wo-agen sys em wi h explici
FQ,ET
...]
8