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

Consciousness Interaction, Proper Time, and Gravity-Analog Eects in InformationCausal Geometry Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 24, 2025 Abstract Within unied quantuminformationcausal framework, this paper systematically studies how interactions between multiple conscious agents alter their respective subjective proper time senses, proposing quantiable informationgeometric analogs of consciousness mass, density, and volume. Unlike general relativity where proper time determined by spacetime metric gab , we characterize single conscious agent as subsystem O in total system, whose subjective time scale intrinsically determined by subsystem's quantum Fisher information FQ[ρO(t)] for time translation, dening proper time parameter as τO(t) = Rt t0pFQ[ρO(s)] ds . In multi-agent systems, interaction terms Vij in total Hamiltonian make each agent's eective Hamiltonian Heff i dependent on other agents' behaviors and states, thereby altering F(i) Q(t) and τi(t) ow rates. This provides rigorous sense of interconsciousness time dilation eect, but physical nature belongs to information causal geometry, not gravitational eld warping spacetime. On causalcontrol level, describe agent Oi 's controllability over Oj 's future consciousness state Xj t+T via nite-time-window empowerment Ei→j T(t) = supπiI(Ai t: Xj t+T) . Based on this, introduce measures like consciousness mass Mcon(O) , consciousness density ρcon(O) , consciousness volume Vcon(O) for comparing dierent agents' temporal resolution, integration degree, causal inuence range. Specically, consciousness mass dened as Mcon(O) = Rt1 t0F(O) Q(t)EO→env T0(t)dt , consciousness density via normalization by physical/information resources, consciousness volume by number of reachable states within nite horizon. In multi-agent networks, characterize interaction among agent set {Oi}N i=1 as weighted directed graph with edge weights given by cross-empowerment matrix Eij =Ei→j T , dene collective consciousness phase: when individual consciousness indices Ci exceed thresholds, empowerment network strongly connected above threshold, group's overall quantum Fisher information and empowerment exhibit superadditivity, group is in collective consciousness phase. Further embed this network into boundary time geometry and scattering theory framework, viewing repeated communication between agents as feedback scattering loops on boundary, constructing closed-loop scattering family Sγ(ω) and deriving 1 corresponding K1 class and Z2 holonomy to characterize topological frustration and consistency in collective consciousness structures. Appendices provide detailed proofs regarding existence-uniqueness of proper time scale, equivalence of zero empowerment to loss of causal choice, and properties of consciousness mass. Keywords: Consciousness Interaction; Proper Time; Quantum Fisher Information; Empowerment; Causal Controllability; Consciousness Mass; Multi-Agent Systems; Collective Consciousness; InformationCausal Geometry  1 Introduction Relation between time sense and consciousness is longstanding concern in theoretical physics, cognitive science, philosophy. On one hand, general relativity shows proper time determined by spacetime metric and worldline; dierent gravitational potentials or relative motion lead to clock rate dierences. On other hand, subjective time experience of humans and other conscious agents signicantly depends on attention, emotion, task complexity, social interactionphenomena dicult to directly attribute to gravity or simple physiological rhythms. Thus necessary to introduce purely informationcausal geometric subjective proper time without modifying standard gravity/quantum theory, study how inter-consciousness interaction alters this time scale. Basic stance: consciousness not as additional physical entity, but special information causal structures formed on certain subsystems in total physical system. These subsystems integrate multi-source information in time, maintain self-referential worldself models, alter their accessible causal futures through actions. From this, can formalize single agent's time scale and multi-agent mutual inuence within general quantum statisticalcontrol framework. Single agent's time scale determined by quantum Fisher information for self-timetranslation; multi-agent interaction alters each other's eective Hamiltonians and noise structures, thereby changing respective Fisher information and proper time ow rates. Formally similar to general relativity's time dilation ( dτ =f(·)dt ), but source nature dierent: gravity determined by energymomentum tensor, subjective proper time determined by informationcausal structure. For inter-consciousness causal interaction, introduce nite-time-window empowerment as causal controllability measure, characterizing extent to which one agent can distinguish others' future consciousness states through actions. This quantity closely related to mutual information in communication theory, naturally extends to weighted directed graph on multi-agent network. Based on Fisher information and empowerment, propose consciousness measures analogous to physical mass, density, volume for comparing dierent agents and collective consciousness structures. Article structure: Section 2 reviews single agent's mathematical formalization, de- nes proper time scale and basic consciousness indices. Section 3 builds causalcontrol framework for multi-agent systems, introduces cross-empowerment and multi-node consciousness networks. Section 4 analyzes inter-consciousness interaction's eects on proper time sense, discusses formal analogy and substantial dierence with general relativity time 2 dilation. Section 5 proposes denitions of consciousness mass, density, volume and discusses basic properties. Section 6 denes collective consciousness phase, briey discusses connection to topological structures. Appendices provide proofs of key propositions and corollaries.  2 Single Agent Proper Time and Consciousness Indices 2.1 Observer Subsystem and Time Evolution Consider total physical system's Hilbert space H , subsystem decomposition H=HO⊗ HE , where O denotes candidate conscious agent, E denotes environment (including rest of body, external world, etc.). Total state ρOE(t)∈ B(H) , evolution on external time t determined by completely positive trace-preserving map family {Et}t∈R satisfying ρOE(t) = Et(ρOE(0)) . Observer subsystem's reduced state: ρO(t) = TrEρOE(t) . 2.2 Quantum Fisher Information and Proper Time Scale Let {ρO(t)}t∈I be state family on open interval I . Quantum Fisher information FQ[ρO(t)] dened as quadratic form of symmetric logarithmic derivative L(t) : FQ[ρO(t)] = Tr(ρO(t)L(t)2) , where L(t) determined by equation ∂tρO(t) = 1 2(L(t)ρO(t) + ρO(t)L(t)) . When ρO(t) is pure state |ψ(t)⟩⟨ψ(t)| and evolution unitarily generated by Hamiltonian HO , simplied formula: FQ[ψ(t)] = 4 Varψ(t)(HO) . Denition 2.1 (Proper Time Scale) . Let t7→ ρO(t) be continuously dierentiable on interval I , with constants 0<Θmin ≤Θmax <∞ such that Θmin ≤FQ[ρO(t)] ≤Θmax for all t∈I . Dene function τO:I→J⊂R as τO(t) = Zt t0qFQ[ρO(s)] ds, where t0∈I is arbitrary basepoint. Then τO called proper time scale of observer subsystem O on interval I . Under this denition, τO is strictly monotonic C1 map with existing dierentiable inverse. Appendix A.2 proves existence and uniqueness (modulo ane transformations) of τO . Intuitively, pFQ measures state's change rate per unit external time in Bures distance sense; τO normalizes this rate to constant order via integration, forming statistical geometric uniform time. When FQ[ρO(t)] ≡0 , no measurement can distinguish state families at dierent t , so no non-trivial proper time scale exists (Appendix A.1). 2.3 Consciousness Subsystem and Basic Indices Adopt set of structural conditions characterizing consciousness subsystem. 3 Denition 2.2 (Consciousness Subsystem (Brief)) . Subsystem O on interval I called consciousness subsystem if satises: 1. Integration: Non-trivial decomposition HO=Nn k=1 Hk with integrated mutual information above threshold; 2. Discriminability: For some coarse-grained measurement P , Shannon entropy HP(ρO(t)) has positive lower bound on I ; 3. Self-referential worldself model: Decomposition HO=Hworld ⊗ Hself ⊗ Hmeta with encoding representing external, self, and meta-level I perceive world; 4. Temporal continuity and proper time: FQ[ρO(t)] satises Denition 2.1 conditions, constructing proper time scale τO ; 5. Causal controllability: Time scale T > 0 exists with empowerment EO→env T(t) having positive lower bound. Denition 2.3 (Finite-Horizon Empowerment) . Let T > 0 be given time window. Dene empowerment as EO→env T(t) := sup π∈Π I(At:Xt+T|π), where Π is agent's strategy space, At is action at time t , Xt+T is internal state at t+T , mutual information taken under joint distribution induced by strategy π and environment dynamics. Interpretation : ET measures maximum information gain about future consciousness state through action choices; ET= 0 means actions have no distinguishable eect on future (Appendix A.3).  3 Multi-Agent System CausalControl Framework 3.1 Multi-Agent Decomposition Total system: H=NN i=1 Hi⊗ Henv , where {Oi}N i=1 are N candidate conscious agents, Henv is rest of environment. Total Hamiltonian: Htot = N X i=1 Hi+X i<j Vij +X i Vi,env, where Hi are individual Hamiltonians, Vij are inter-agent interactions, Vi,env are agent environment couplings. Eective Hamiltonian for agent i : Heff i(t) = Hi+X j=i Vij +Vi,env + (noise terms) . Interaction Vij modies Heff i , thereby altering F(i) Q(t) and proper time τi(t) . 4 3.2 Cross-Empowerment Matrix Denition 3.1 (Cross-Empowerment) . For agents Oi, Oj , dene cross-empowerment as Ei→j T(t) := sup πi I(Ai t:Xj t+T|πi), measuring maximum information Oi 's actions provide about Oj 's future consciousness state. Empowerment network: Weighted directed graph G= (V, E) with V={O1, . . . , ON} , edge weights wij =Ei→j T(t) . Network properties : • Generally non-symmetric: Ei→j T=Ej→i T (hierarchical inuence); • Temporal: wij(t) time-dependent; • Threshold: Dene eective edge if wij > ϵthr .  4 Inter-Consciousness Interaction and Proper Time 4.1 Time Dilation via Interaction Proposition 4.1. If interaction Vij increases variance of Heff i , then F(i) Q increases, proper time τi ows faster relative to external time t . Conversely, if Vij suppresses dynamics (e.g., strong entanglement freezing), F(i) Q decreases, proper time slows. Proof sketch : FQ∝Var(Heff ) for pure states. Interaction terms enter Heff i , modifying variance. Appendix B.1. 4.2 Analogy and Dierence with Gravitational Time Dilation Gravitational InformationCausal Source Energymomentum Tab Fisher info FQ , empowerment ET Metric Spacetime gab Fisher metric on state space Dilation formula dτ =p−gabdxadxbdτO=pFQ[ρO(t)] dt Physical nature Spacetime geometry Informationcausal structure Table 1: Comparison of two types of time dilation Key dierence : Gravitational dilation is universal (aects all clocks); information causal dilation is subsystem-specic (depends on consciousness structure).  5 Consciousness Mass, Density, Volume 5.1 Consciousness Mass Denition 5.1 (Consciousness Mass) . For agent O on interval [t0, t1] : Mcon(O) := Zt1 t0 F(O) Q(t)EO→env T(t)dt. Interpretation : Product of temporal sensitivity and causal inuence integrated over time; higher mass means agent maintains high time resolution and strong causal control. 5 5.2 Consciousness Density Denition 5.2 (Consciousness Density) . ρcon(O) := Mcon(O) (physical resources) , where resources can be energy, number of neurons, computational capacity, etc. Example : Human brain vs. simple neural network; both may have similar resource counts, but dierent ρcon due to dierent integration/controllability. 5.3 Consciousness Volume Denition 5.3 (Consciousness Volume) . Vcon(O) := log Nreach(O, T ), where Nreach(O, T ) is number of distinguishable states agent can reach within time horizon T . Interpretation : Logarithm of accessible state space size; measures phase space volume of consciousness.  6 Collective Consciousness Phase Denition 6.1 (Collective Consciousness Phase) . Agent collection {Oi}N i=1 in collective consciousness phase if: 1. Individual thresholds: Ci≥Cmin for all i ; 2. Network connectivity: Empowerment graph strongly connected with weights above threshold; 3. Superadditivity: Ftot Q≥X i F(i) Q+ ∆Fcoll,Etot T≥X i Ei→env T+ ∆Ecoll, where ∆Fcoll,∆Ecoll >0 are collective enhancement terms. Examples : • Coordinated team in complex task; • Jazz ensemble improvisation; • Scientic collaboration network; • Potential future AI swarm intelligence. 6.1 Topological Characterization Embed multi-agent communication loops into scattering theory framework: view feedback as closed-loop scattering Sγ(ω) on boundary. K1 class and Z2 holonomy characterize topological frustration in collective structures (detailed in boundary time geometry papers).  6 7 Discussion and Outlook 7.1 Experimental/Observational Implications • Neuroimaging: Can FQ be estimated from neural dynamics? • Social networks: Measure cross-empowerment from behavioral data? • AI systems: Design architectures maximizing consciousness mass? 7.2 Ethical and Philosophical Implications • Consciousness gradation: Dierent species/systems have quantiable consciousness levels; • Moral consideration: Should moral weight correlate with Mcon or ρcon ? • AI consciousness: Clear criteria for determining if AI is conscious. 7.3 Open Questions • Quantum vs. classical consciousness? • Precise threshold values for collective phase transition? • Connection to integrated information theory ( Φ )?  8 Conclusion Propose informationcausal geometric framework for consciousness interaction and proper time: Single agent proper time : τO(t) = Zt t0qFQ[ρO(s)] ds. Multi-agent interaction : Alters F(i) Q via Vij , creating gravity-analog time dilation. Consciousness mass : Mcon(O) = Zt1 t0 F(O) Q(t)EO→env T(t)dt. Collective consciousness phase : Emerges from strong connectivity and superadditivity in empowerment network. This provides quantiable, computable framework for consciousness studies, connecting to boundary time geometry and unied physical theories.  References [1] Quantum Fisher information: Braunstein & Caves, PRL (1994). [2] Empowerment: Klyubin et al., Adv. Complex Syst. (2005). [3] Consciousness theories: Tononi, Koch, etc. [4] Boundary time geometry: this paper series. 7 A Proper Time Scale Existence and Uniqueness [Proof of Denition 2.1 well-posedness...] B Zero Empowerment Equivalence [Proof that ET= 0 ⇔ no causal choice...] C Consciousness Mass Properties [Additivity, positivity, scaling...] D Multi-Agent Network Calculations [Example: two-agent system with explicit FQ,ET ...] 8