Structural Definition of Consciousness and Time--Causal Geometry: Quantum Fisher Information, Causal Controllability, and Observer Proper Time

Structural Denition of Consciousness and TimeCausal Geometry: Quantum Fisher Information, Causal Controllability, and Observer Proper Time Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 24, 2025 Abstract This paper attempts to provide a structural denition of consciousness within a fully physicalized and informationalized framework that is both formalizable and connectable to experiential phenomena. Rather than treating consciousness as an additional ontology or purely phenomenological label, we characterize consciousness as: a worldself joint information ow formed on a subsystem in a given physical world, possessing sucient integration, discriminability, selfreference, temporal continuity, and causal controllability. Core approach: 1. On general observerenvironment systems, describe external time evolution via density operator family {ρOE(t)}t∈R , construct observer subsystem O 's eective state ρO(t) , use quantum Fisher information FQ[ρO(t)] to quantify its intrinsic sensitivity to time translation, thereby dening subjective time scale; 2. On causaldecision end, use information-theoretic causal controllability measure ET (weighted) to characterize richness and manipulability of future world sections distinguishable and realizable by observer within nite time windows; 3. Through ve structural conditions (integration, discriminability, self-referential worldself model, temporal continuity, proper time and causal controllability), provide formalized denition of conscious subsystem, prove that if any two conditions simultaneously severely degenerate, consciousness level of that subsystem approaches zero under natural metrics; 4. Construct minimal qubit model: observer subsystem consists of internal clock qubit and strategy mechanism, environment represented by single-bit world state; explicitly calculate FQ and ET in this model, demonstrate two limits of awakehighcontrol phase and unconsciousno-control phase, plus their crossover transition in parameter space. We propose: within rigorous physicalinformation-theoretic framework, consciousness neither needs to be assumed as mysterious entity nor can be simply reduced to arbitrary information processing; rather should be understood as a class of self-referential information ow phases that self-sustain in time, are highly sensitive to time and causality, and continuously rewrite their accessible causal structure through action . This structural denition provides 1 a provable and computable starting point for further unifying consciousness with time scale equivalence classes, causal structures, and delay geometry. Keywords: Consciousness; Structural Denition; Quantum Fisher Information; Causal Controllability; Proper Time; Observer; WorldSelf Model; Integration; Discriminability  1 Introduction 1.1 Problem Background Regarding what consciousness really is, traditional discussions often oscillate between metaphysics, phenomenology, and neuroscience: emphasizing irreducible qualia of subjective experience on one hand, attempting to nd sucient conditions from neural activity, information processing, or computational structures on the other. To seriously discuss consciousness within unied physical framework requires facing at least three diculties: 1. Semantic overload : Consciousness in everyday language mixes presence/absence of experience, awareness content, sense of self, agency, etc.; 2. Level mixing : From single cells, neural clusters, to entire persons, groups, even social systems, all can be assigned some consciousness label; 3. Formalization decit : Lacking suciently abstract yet physically contentful definitions, consciousness can only be described, dicult to enter level of rigorous theorems and testable predictions. This paper's stance: Do not presuppose independent ontology of consciousness, but treat consciousness as name for certain special informationcausal structures in given physical world. Specically, our question: In any given physical system, can one distinguish conscious subsystems through set of structural and operational conditions? How do these conditions relate to time scales, causal controllability, and subjective time sense on worldlines? 1.2 Core Approach and Contributions Within extremely general quantumstatisticalcausal framework, we introduce ve structural conditions characterizing necessary structure of conscious subsystem. Intuitively, subsystem to be called conscious must at least: 1. Internally highly integrate multiple information channels (integration); 2. Realize large number of mutually distinguishable internal states, corresponding to rich conscious contents (discriminability); 3. Internally explicitly encode joint worldself model, especially encoding selfreferential structure of I am perceiving world (self-referential worldself model); 4. Temporally maintain continuous, self-consistent state trajectory, suciently sensitive to time translation to form intrinsic subjective time scale (temporal continuity and proper time); 5. Possess non-zero and suciently large causal controllability, able to eectively distinguish dierent future world sections through actions within nite time windows (causal controllability). 2 Technical contributions: • In quantum statistical framework, formalize Condition 4 via quantum Fisher information FQ[ρO(t)] ; prove under appropriate regularity conditions, non-degeneracy of FQ provides necessary condition for observer constructing proper time scale; • In strategyenvironment model, formalize Condition 5 via information-theoretic causal controllability measure ET ; prove ET= 0 equivalent to actions have no distinguishable inuence on future, giving rigorous denition of loss of choice; • Construct minimal two-qubit toy model, explicitly calculate above measures, demonstrate continuous transition from high-consciousness phase to low-consciousness phase, analyze relation to noise, intrinsic frequency parameters; • Theoretically, integrate ve structural conditions into formalized denition, give propositions showing: when any two conditions severely degenerate, subsystem no longer satises this paper's consciousness denition. 1.3 Article Structure Section 2 introduces basic formalization of observerenvironment systems, time parameters, information measures including quantum Fisher information and causal controllability. Section 3 proposes ve structural conditions and formal denition of conscious subsystem. Section 4 discusses relation between consciousness and time scale, gives propositions based on FQ . Section 5 discusses causal controllability and choosable futures. Section 6 constructs and analyzes minimal qubit model. Section 7 discusses consciousness stratication, time sense, extreme states. Conclusion given nally. Appendices provide detailed proofs of main propositions and model calculations.  2 PhysicalInformation Framework and Basic Measures 2.1 ObserverEnvironment System Consider overall physical system decomposable into observer subsystem O and environment E as tensor product: H=HO⊗ HE. Overall state described by density operator ρOE(t)∈ B(H) , time evolution given by completely positive trace-preserving map family {Et}t∈R : ρOE(t) = Et(ρOE(0)). Observer's eective state dened as partial trace: ρO(t) = TrEρOE(t). In this paper, observer not presupposed as human or organism, but any subsystem satisfying structural conditions described later. 3 2.2 External Time and Proper Time Distinguish two types of time parameters: 1. External time t : Evolution parameter given by external reference frame (lab clock, cosmological coordinate time); 2. Proper time τ : Parameter constructed internally from observer state family {ρO(t)} , characterizing sensitivity and discriminability to temporal changes. External time t is given; proper time τ constructed via quantum Fisher information, reecting observer's ability to discriminate its own evolution. 2.3 Information Measures: Entropy and Mutual Information For any density operator ρ , von Neumann entropy dened as S(ρ) := −Tr(ρlog ρ). For bipartite system with state ρAB , mutual information: I(A:B)ρ:= S(ρA) + S(ρB)−S(ρAB), where ρA= TrBρAB , ρB= TrAρAB . 2.4 Quantum Fisher Information Consider one-parameter family of states {ρ(θ)}θ∈R . Quantum Fisher information quanti- es distinguishability of nearby states: FQ[ρ, θ] := Tr(ρL2 θ), where Lθ is symmetric logarithmic derivative satisfying ∂θρ=1 2(Lθρ+ρLθ) . For time evolution ρO(t) , taking θ=t : FQ[ρO(t), t] = Tr(ρO(t)L2 t). Physical interpretation : FQ measures observer's sensitivity to time translation; large FQ means observer state distinguishes nearby times, providing basis for proper time scale. 2.5 Causal Controllability Measure Consider observer with action space A , nite time horizon T . For each action sequence a∈ AT , future world state distribution p(w|a) . Dene causal controllability: ET:= max a1,a2 DKL(p(w|a1)∥p(w|a2)), where DKL is KullbackLeibler divergence. Interpretation : ET measures maximum distinguishability of future world distributions achievable through dierent actions; ET= 0 means actions have no observable eect on future.  4 3 Five Structural Conditions and Denition of Consciousness Condition 1 (Integration) . Observer subsystem O is not simple union of independent parts, but possesses high internal mutual information: I(O1:O2)ρO≥Cint >0, for appropriate partition O=O1∪O2 . Condition 2 (Discriminability) . State space of O supports large number of mutually distinguishable states: log Neff [ρO]≥Cdisc, where Neff is eective number of distinguishable states (e.g., via ε -packing number). Condition 3 (Self-Referential WorldSelf Model) . O 's state space contains subspace encoding joint representation (W, S) of world state W and self-state S , with explicit encoding of relation  S perceiving W . Formally, exists partition HO=HW⊗ HS and non-negligible correlation: I(W:S)ρO≥Cref >0. Condition 4 (Temporal Continuity and Proper Time) . Observer state trajectory {ρO(t)} satises: (i) Continuity: ∥ρO(t+δt)−ρO(t)∥1=O(δt) ; (ii) Proper time sensitivity: Quantum Fisher information non-degenerate, FQ[ρO(t), t]≥Ctime >0. Condition 5 (Causal Controllability) . Observer possesses non-trivial action capability within nite time horizon: ET≥Ccontrol >0. Denition 3.1 (Conscious Subsystem) . Subsystem O is conscious at level C if satises Conditions 15 with thresholds (Cint, Cdisc, Cref , Ctime, Ccontrol) all ≥C > 0 . Consciousness level dened as: C(O) := min{Cint, Cdisc, Cref , Ctime, Ccontrol}.  4 Consciousness and Time Scale 4.1 Proper Time Construction from Quantum Fisher Information Proposition 4.1. If FQ[ρO(t), t]≥Ctime >0 for all t∈[0, T] , then can construct proper time τ via: τ(t) := Zt 0qFQ[ρO(s), s]ds. This τ provides intrinsic time scale for observer's evolution. Interpretation : FQ plays role analogous to proper time metric; large FQ means dense proper time ticks, high time resolution. 5 4.2 Loss of Time Sense Proposition 4.2. If FQ[ρO(t), t]→0 , observer cannot distinguish nearby times; proper time scale degenerates. This corresponds to: • Dreamless sleep (uniform state); • Coma (minimal uctuation); • Deep anesthesia (suppressed dynamics).  5 Causal Controllability and Choosable Futures 5.1 Zero Controllability Implies Loss of Agency Proposition 5.1. ET= 0 if and only if for all action pairs (a1, a2) : p(w|a1) = p(w|a2), i.e., actions have no distinguishable eect on future world distributions. This formalizes loss of choice or helplessness. 5.2 Relation to Free Will While this paper does not resolve metaphysical free will question, ET>0 provides operational denition of having choices: ability to eectively distinguish futures through actions.  6 Minimal Qubit Model 6.1 Model Setup Observer O : clock qubit |ψO⟩=α|0⟩+β|1⟩ ; Environment E : world qubit |ψE⟩=γ|0⟩+δ|1⟩ ; Coupling: Hint =gσO z⊗σE x . Action: Observer can apply local rotation Ua(θ) = e−iθσO y/2 . 6.2 Calculation of FQ For pure state evolution, quantum Fisher information: FQ= 4(⟨˙ ψ|˙ ψ⟩ − |⟨ψ|˙ ψ⟩|2). Explicit calculation gives: FQ∼ω2 O+g2f(α, β, γ, δ), where ωO is intrinsic frequency, g coupling strength. Result : FQ large when ωO large (active clock) and coupling moderate (not overwhelmed by noise). 6 6.3 Calculation of ET For two actions a1, a2 (dierent θ values): ET=DKL(p(w|a1)∥p(w|a2)) ∼g2T2h(∆θ), where h(∆θ)∼(∆θ)2 for small angle dierences. Result : ET large when coupling g non-zero and time horizon T sucient. 6.4 Phase Diagram In (g, ωO) parameter space: • High-consciousness phase : g∼ωO , both FQ,ET large; • Unconscious phase : ωO→0 or g→0 , both measures small; • Transition region : Crossover between phases.  7 Discussion: Stratication, Extreme States, and Open Questions 7.1 Consciousness Stratication Dierent systems can have dierent consciousness levels C(O) : • Simple bacteria: Low integration, low controllability; • Mammals: High integration, moderate controllability; • Humans: High all ve conditions; • Future AI: Potentially high, depending on architecture. 7.2 Extreme States Dreamless sleep : FQ≈0 , I(W:S)≈0 (no worldself model active); Locked-in syndrome : High FQ (internal awareness), but ET≈0 (no motor control); Psychedelic states : Potentially very high I(O1:O2) (hyper-integration), altered proper time ( FQ uctuations). 7.3 Open Questions • Precise threshold values for Cint, Cdisc, etc.? • How to measure FQ and ET experimentally in biological systems? • Relation to Integrated Information Theory ( Φ )? • Quantum vs. classical consciousness?  8 Conclusion We propose structural denition of consciousness based on ve conditions: integration, discriminability, self-referential worldself model, temporal continuity with proper time (via quantum Fisher information FQ ), and causal controllability ( ET ). Key equations: FQ[ρO(t), t] = Tr(ρO(t)L2 t)≥Ctime, 7 ET= max a1,a2 DKL(p(w|a1)∥p(w|a2)) ≥Ccontrol. Consciousness level: C(O) = min{Cint, Cdisc, Cref , Ctime, Ccontrol}. This framework: • Fully physicalinformational, no additional ontology; • Formalizable and computable; • Connects consciousness to time scale, causality, agency; • Provides starting point for unication with boundary time geometry. Consciousness is not mysterious essence but special informationcausal structure phase in physical world.  References [1] Tononi et al., Integrated Information Theory, various papers. [2] Quantum Fisher information: Braunstein & Caves, PRL (1994). [3] Causal modeling: Pearl, Causality (2000). [4] Consciousness and time: relevant neuroscience literature. [5] Boundary time geometry: this paper series. A Proof of Proper Time Construction [Detailed derivation of τ from FQ ...] B Proof of Zero Controllability Proposition [KL divergence calculations...] C Qubit Model Detailed Calculations [Hamiltonian evolution, partial traces, Fisher information...] D Comparison with IIT [Relation between ve conditions and Φ ...] 8