Unified Mathematical Definition of ``Self'': Causal Manifold, Observer, and Self-Referential Scattering Network

Unied Mathematical Denition of Self: Causal Manifold, Observer, and Self-Referential Scattering Network Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract Within the framework of unied time scale, causal manifolds, and boundary time geometry, we provide an axiomatic mathematical denition of the rst-person subject self. The basic conception is: self is not a label for an instantaneous physical conguration, but rather an equivalence class of self-referential observer structures along a timelike worldline ordered by the unied time scale. At the geometric level, the universe is modeled as a globally hyperbolic Lorentzian manifold with causal partial order, where generalized entropy extremization and quantum energy conditions on small causal diamonds yield gravitational eld equations and time arrow. At the spectral and scattering level, the unied time scale serves as the mother scale via the scale identity κ(ω) = φ′(ω)/π =ρrel(ω) = (2π)−1tr Q(ω) , unifying phase gradient, relative density of states, and WignerSmith time-delay trace as a single temporal density. At the causal network and information geometry level, observers are formalized as multi-component objects with local causal domain, resolution scale, boundary observable algebra, state, model family, and update operator; multiple observers form consensus geometry via communication channels and relative entropy contraction. At the self-referential scattering and consciousness level, consciousness is characterized as a two-port scattering network with delay kernel and time delay, where self-feedback closed loops exhibit a Z2 holonomy of modied determinant square root, manifesting the topological invariant of self. Building on this foundation, we introduce the Istructure: a timelike worldline γ with unied time scale, equipped with internal observable algebra, time-stamped state family, selfreferential update operator, memory subsystem, and internal environment maps, satisfying causal locality, persistent and distinguishable internal memory, explicit dependence of updates on self's future behavior and environment predictions, and compatibility with generalized entropy dynamics. After dening an equivalence relation in terms of time rescaling and algebra embeddings, we regard equivalence classes as individual selves. We prove that under appropriate causal and energy conditions, each Istructure equivalence class corresponds to a minimal strongly connected self-referential scattering closed loop in the causal network, whose unied time scale and generalized entropy gradient align monotonically; conversely, each minimal selfreferential scattering closed loop satisfying this alignment condition denes a unique Istructure equivalence class. In the appendices, we provide an outline of existence and uniqueness proofs for constructing I-worldlines from local observer data, and the construction rewriting I-structures as closed loop scattering families with Z2 holonomy, showing compatibility of its mod-two indicator with fermionic commutation phase and topological class under NullModular double cover. Thus, self is characterized as a unied mathematical object with geometric support, information kernel, and topological ngerprint. Keywords Causal manifold; Unied time scale; Observer; Consciousness; Self-referential scattering network; Z2 holonomy; Generalized entropy 1 1 Introduction & Historical Context In the standard framework of relativity and quantum theory, observer is often treated as a passive reference frame or measurement device, while the rst-person subject question who am I is left to philosophy. To rigorously answer this question at the physical and mathematical level requires a unied description centered on causal structure, time scale, and information geometry, embedding concepts such as subject, time, memory, and self-reference into the same geometric and operator language. On one hand, scattering theory and spectral theory show that time can be viewed as the derivative of scattering phase and function of density of states. For Schrödinger operators with relative trace-class perturbations, the BirmanKren formula gives the relationship between scattering determinant and spectral shift function det S(λ) = exp(−2πiξ(λ)) , whose derivative ξ′(λ) is interpreted as relative density of states, further linked to the trace of WignerSmith time-delay operator Q(ω) = −iS(ω)†∂ωS(ω) . Through this route, one can view time scale as the unied invariant of φ′(ω)/π , spectral shift derivative −ξ′(ω) , and time-delay trace (2π)−1tr Q(ω) . Related results can be found in the original work of BirmanKren and subsequent systematic studies of scattering phase and determinant. The time delay matrix proposed by Wigner and Smith has been systematically developed in multi-channel scattering, disordered media, electromagnetic and acoustic scattering. On the other hand, at the intersection of algebraic quantum eld theory and gravity, Tomita Takesaki modular theory reveals that for any von Neumann algebra with a suciently faithful state, there exists a modular ow uniquely determined by the state, allowing one to extract a oneparameter group from the timeless algebraic structure. This structure was proposed by Connes and Rovelli as the foundation for the thermal time hypothesis: physical time ow is not universal, but determined by the modular ow of a given statistical state. This provides an operator-algebraic perspective on the intrinsic nature of time. At the boundary of gravity and quantum information, the variation of generalized entropy Sgen and quantum energy conditions provide a route to derive Einstein equations from entropy balance. Work by Jacobson, Faulkner, and others shows that under appropriate semi-classical and holographic assumptions, generalized entropy extremization and second-order non-negativity along small causal diamond boundaries are equivalent to Einstein equations with cosmological constant. Proofs of the quantum null energy condition (QNEC) and averaged null energy condition (ANEC) further strengthen the equivalence structure among entropyenergygeometry. This direction shows that local causal structure can be viewed as the macroscopic manifestation of generalized entropy optimization principles. The above scatteringspectral theory, modular theory, and entropygeometry theory provide the foundation for constructing unied time scale and causal manifolds. Building on this, one can view the universe as a causal manifold under unied time scale, whose boundary carries observable algebra and state, with boundary time geometry gluing scattering phase, modular time, and gravitational boundary terms into the same structure. Meanwhile, information geometry and statistical causal inference frameworks show that multi-observer systems can be characterized via relative entropy and Fisher metric, capturing model update and consensus formation. Regarding consciousness and the subject problem, much work has been devoted to constructing information-theoretic or physicalist descriptions, such as viewing consciousness as specic integrated information structure, global workspace, or multi-layer encoding process. However, these theories typically lack rigorous geometricoperator formalism compatible with causal manifolds, unied time scale, and quantum gravity. On the other hand, topological or algebraic descriptions of selfreference and self are relatively scattered. This paper, within the framework of unied time scale and causal manifolds, attempts to provide 2 a mathematical object for self: a timelike worldline with time scale, equipped with internal observable algebra, state family, self-referential update operator, and memory structure, satisfying causal locality and generalized entropy consistency, and rewritable in self-referential scattering networks as a minimal strongly connected closed loop with Z2 holonomy. By constructing an equivalence relation, we understand equivalence classes of such structures as dierent realizations of the same self, thereby formalizing the subject in a rigorous mathematical context. 2 Model & Assumptions This section provides the basic structure of causal manifold, unied time scale, observer, and selfreferential scattering network, along with adopted assumptions. 2.1 Causal Manifold and Small Causal Diamonds Assume the universe at large scales is described by a four-dimensional Lorentzian manifold (M, g) satisfying: 1. Global hyperbolicity : There exists a Cauchy surface Σ such that every non-spacelike curve intersects Σ exactly once. 2. Stable causality : No closed timelike curves exist, and small perturbations do not produce causal violations. 3. Causal partial order : Use p≺q to denote the existence of a future-directed timelike or null curve from p to q . For any p∈M and suciently small positive r , take p± as points obtained by evolving along some future-directed timelike geodesic with proper time parameter ±r , and dene the small causal diamond Dp,r =J+(p−)∩J−(p+). Its boundary is generated by two families of null geodesics, providing the basic structure for analyzing local gravitational eld equations and generalized entropy changes. Assume there exists appropriate quantum eld theory coupled to gravity such that for each small causal diamond boundary, one can dene generalized entropy Sgen =Area(∂Dp,r) 4Gℏ+Sout, where Sout is the von Neumann entropy of external eld degrees of freedom. Assume QNEC and related entropyenergy inequalities hold and can be used to equivalently characterize local gravitational eld equations. 2.2 Unied Time Scale and Mother Identity In scattering and spectral theory, consider a pair of self-adjoint operators (H0, H) , where H is a trace-class or relative trace-class perturbation of H0 . Let S(ω) be the scattering matrix at energy ω , ξ(ω) the spectral shift function, with BirmanKren formula giving det S(ω) = exp−2πiξ(ω), 3 yielding relative density of states ρrel(ω) = −ξ′(ω). The WignerSmith time-delay operator is dened as Q(ω) = −iS(ω)†∂ωS(ω), whose trace characterizes the time delay averaged over all channels. Introducing the total scattering half-phase φ(ω) = 1 2arg det S(ω) = −π ξ(ω), we have φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). Scale identity denes the unied time scale density κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), interpreting κ(ω) dω as the eective time scale per unit energy bandwidth. Through geometric optics limit and eikonal approximation, this scale can be linked to gravitational time delay, redshift, and proper time; through modular theory and thermal time hypothesis, it can be aligned with modular time parameter on boundary algebras. In this paper, the unied time scale equivalence class [τ] denotes all time function families compatible with κ(ω) , locally rewritable from scattering phase, modular ow, or geometric time, diering only by ane rescaling and coordinate choice. 2.3 Observer as Structured Causal Agent From the causal network perspective, we introduce abstract observer structure. Denition 1 (Observer) . An observer Oi is dened as a multi-component object Oi= (Ci,≺i,Λi,Ai, ωi,Mi, Ui, ui,(Cij)j), where: 1. Ci⊂M is the reachable causal domain; 2. ≺i is the local causal partial order on Ci ; 3. Λi is the resolution scale (e.g., timeenergy or spacemomentum windows), determining distinguishable frequency bands and spatial scales; 4. Ai is the boundary observable C∗ algebra associated with Oi ; 5. ωi is a state on Ai ; 6. Mi is a family of candidate causal dynamical models; 7. Ui is an update operator based on observational data and models (generally completely positive trace-preserving maps or Bayesian update operators); 8. ui is a utility function or decision preference; 4 9. Cij are communication channels with other observers Oj (completely positive trace-preserving maps or classical channels). On the common observable algebra Acom =\ i Ai, let the multi-observer state family (ω(t) i) update with discrete or continuous time. If the communication graph is strongly connected and there exists a common xed point ω∗ , then the weighted relative entropy Φ(t)=X i λiD(ω(t) i∥ω∗) constitutes a Lyapunov function for appropriate update rules, ensuring formation of state consensus. This structure provides the foundation for causal and informational interaction among multiple selves. 2.4 Internal Algebra, Memory and Self-Referential Update To characterize the internal persistence and memory of a subject, we need to extract subalgebras and update structures within a single observer. Denition 2 (Internal Structure) . Given a timelike trajectory γ:I→M with τ(γ(t)) = t , its internal structure is a triple (Aint,(ωint t)t∈I, U), where: 1. Aint is the C∗ algebra of internal degrees of freedom; 2. For each t∈I , ωint t is a state on Aint ; 3. For any t2> t1 , there exists a completely positive trace-preserving map U(t2, t1) : Aint → Aint, satisfying the semigroup condition U(t3, t2)◦U(t2, t1) = U(t3, t1), ωint t2=ωint t1◦U(t2, t1). Denition 3 (Memory Subsystem) . A commutative subalgebra C ⊂ Aint is called a memory subsystem if: 1. C is ∗ -isomorphic to bounded functions on some measure space or nite-dimensional diagonal matrix algebra; 2. The probability measure family µt induced by ωint t on C forms a Markov process; 3. For any t2> t1 , there exist measurable sets such that the dependence of µt2 on µt1 cannot be eliminated by external environment variables, ensuring memory carries genuine information about subsequential observations. 5 The existence of memory subsystem ensures the subject has a traceable internal history over time, rather than complete reset at each moment. Subject self-referentiality requires explicit dependence of updates on internal state and environment model. To this end, we introduce internal environment maps. For each t , let Aext t be the external observable algebra accessible to the subject around γ(t) , and dene a completely positive map Et:Aext t→ Aint, characterizing the encoding of external world within the subject's interior. Denition 4 (Self-Referential Update) . If there exists a functional F such that for any t2> t1 , U(t2, t1) = Ft2, t1;ωint t1, Et1,Dext [t1,t2], where Dext [t1,t2] denotes external observational data available in the interval, then the update is called self-referential. Here ωint t1 via Et1 provides internal prediction of future environment states, inuencing its own evolution. 2.5 Self-Referential Scattering Networks and Z2 Holonomy In the scattering description, complex systems and their environments can be represented as networks formed by node scattering matrices Sj(ω) interconnected via waveguides, delay lines, and feedback. For a given topological structure and parameter family, one uses Redheer star product to construct the closed-loop scattering matrix S⟲(ω;λ) , where λ represents slowly varying control parameters (such as internal strategies, attention, or external conditions). Under trace-class or relative trace-class conditions, one can dene the modied determinant detpS⟲ and dene the phase index map s(ω, λ) = det pS⟲(ω;λ). Along a closed path γ⊂X◦ in parameter space avoiding singularity sets, the holonomy of the square-root determinant is dened as ν√S⟲(γ) = expiIγ 1 2is−1ds∈ {±1}, giving a Z2 index that is homotopy invariant. This index is associated with the mod-two part of spectral ow, and in many cases can be interpreted as the topological sign of fermionicity. This paper assumes that the self-referential update of a subject can be rewritten as some closedloop scattering network under appropriate frequency domain and inputoutput models, thereby allowing the above Z2 topological ngerprint to be dened on the network. 3 Main Results (Theorems and Alignments) Under the above model and assumptions, this section provides the formal denition of Istructure and main results. 6 3.1 Denition of IStructure Choose a representative τ:M→R from the unied time scale equivalence class [τ] as the time function, and consider a future-directed timelike curve γ:I→M satisfying τ(γ(t)) = t . Denition 5 (Observer Trajectory) . A γ satisfying the above conditions is called an observer trajectory. Denition 6 (IStructure) . On the unied time scale equivalence class [τ] , an Istructure is data I=γ, Aint,(ωint t)t∈I, U, C,(Et)t∈I, satisfying: 1. γ is an observer trajectory; 2. (Aint,(ωint t), U) is an internal structure satisfying Denition 2; 3. C ⊂ Aint is a memory subsystem satisfying Denition 3; 4. Et:Aext t→ Aint are internal environment maps making U self-referential with respect to (ωint t1, Et1) (Denition 4); 5. Causal locality : For each t∈I , there exists a bounded causal domain Kt⊂M such that the inuence of internal state ωint t on external observables is supported in Kt∩J−(γ(t)) ; 6. Entropy consistency : Along the small causal diamond family Dγ(t),r on γ , under state ωint t⊗ωext t , the extremization and second-order non-negativity of generalized entropy Sgen is compatible with gravitational eld equations and QNEC/QFC type constraints. Intuitively, an Istructure is a self-referential observer on a worldline with unied time scale, having persistent memory and causal consistent interaction with the external world. 3.2 Equivalence Relation: Same I in Dierent Realizations Dierent physical realizations (such as dierent bases, time rescalings, hardware) may correspond to the same self. To this end, we need an equivalence relation. Denition 7 (Equivalence of IStructures) . Two Istructures I= (γ, Aint,(ωint t), U, C,(Et)),I′= (γ′,A′int,(ω′ t′int), U′,C′,(E′ t′)) are called equivalent, denoted I∼I′ , if there exist: 1. A strictly monotone bijection f:I→I′ ; 2. A ∗ -isomorphism Φ : Aint → A′int with Φ(C) = C′ , being a measure isomorphism on spectral spaces; 3. For all t∈I , ω′ f(t)int =ωint t◦Φ−1, U′(f(t2), f(t1)) ◦Φ=Φ◦U(t2, t1); 4. External algebra isomorphisms Ψt:Aext t→ A′ f(t)ext such that E′ f(t)= Φ ◦Et◦Ψ−1 t. Denition 8 (Mathematical Object of Self) . A self is dened as an equivalence class of some Istructure [I] = {I′:I′∼I}. 7 3.3 Structural Theorems Based on the above denitions, there are three core results. Theorem 9 (Existence and Local Uniqueness of I-Worldline) . On a globally hyperbolic Lorentzian manifold (M, g) , assume there exists a local observer O∗ whose records satisfy: 1. Under some unied time scale representative τ , a timelike curve γ∗ can be reconstructed from observational records such that the associated observational domain has small causal diamond near-Minkowski structure around γ∗ ; 2. There exists a decomposition A∗≃ Aint ⊗ Aext , and on Aint there exists a stable memory subsystem. Then under appropriate technical assumptions (including QNEC, Hadamard states, nite energy conditions), one can construct an Istructure I∗ along γ∗ , and given the unied time scale equivalence class and internal algebra equivalence class, its equivalence class [I∗] is locally unique. Theorem 10 (Correspondence between IStructures and Minimal Strongly Connected Self-Referential Scattering Closed Loops) . Under the framework of unied time scale and scatteringdelay networks, assume: 1. All associated scattering operators satisfy trace-class or relative trace-class conditions; 2. The internal update and environment maps of Istructure can be rewritten in frequency domain as closed-loop scattering matrix family S⟲(ω;λ) . Dene I-closed loop as minimal strongly connected components satisfying memory, self-referentiality, and topological nontriviality conditions. Then there exists a natural correspondence [I]←→ S([I]), such that each Istructure equivalence class corresponds to a unique I-closed loop and vice versa, with unied time scales and delay spectra on both sides consistent. Theorem 11 (Topological Fingerprint and Z2 Invariant) . For each I-closed loop S , via the modied determinant square root of closed-loop scattering matrix S⟲(ω;λ) , dene a Z2 index ν(S)∈ {±1}, which is invariant under parameter homotopy and Istructure equivalence relation. For appropriate systems, changes in this index are compatible with the mod-two spectral ow of fermionic statistics, topological class in the NullModular double cover, and BF-type Z2 bulk integrals. 4 Proofs This section provides the proof framework and key steps for the main theorems. Complete technical details are expanded in appendices. 8 4.1 Proof of Theorem 1 Step 1: Reconstruct timelike trajectory from local observational data For local observer O∗ , extract spacetime events and their causal relations from its records. Through maximal-volume waist surface or minimal curvature criterion, select representative τ in the unied time scale equivalence class [τ] , and using τ as time function, embed O∗ 's records into the foliation structure {τ−1(t)} . On each time slice, select a center of mass point γ∗(t) , obtaining a timelike curve γ∗ . Global hyperbolicity ensures γ∗ can be chosen as a smooth timelike curve. Step 2: Construct internal algebra and memory subsystem Using the decomposition A∗≃ Aint ⊗ Aext , select from the commutative subalgebra of Aint a subalgebra C containing readable record bits, whose random process on the spectral space is determined by observational records. Conrm via relative entropy Hessian that these degrees of freedom have stable Fisher distinguishability, and their time evolution can be described by a Markov process, thereby satisfying memory subsystem conditions. Step 3: Dene update operator and internal environment maps Using O∗ 's model family M∗ and update rule U∗ , construct internal update U(t2, t1) and internal environment maps Et . These maps are concretely realized through the process of external measurement results → internal memory state. The dependence of updates on internal models and memory guarantees self-referentiality. Step 4: Check causal locality and entropy consistency Around each point of γ∗ , construct small causal diamonds Dγ∗(t),r . Using local near-Minkowski and energy-bounded conditions, apply JacobsonFaulkner type arguments: under state ωint t⊗ωext t , local generalized entropy extremization and second-order non-negativity are equivalent to Einstein equations and QNEC. Thus I∗ does not break existing geometricentropy structure. Step 5: Local uniqueness If another realization I′ ∗ satises the same observational records and unied time scale equivalence class, construct time rescaling f and algebra ∗ -isomorphism Φ making them completely consistent on memory subsystem and observational distributions, hence I∗∼I′ ∗ . 4.2 Proof of Theorem 2 Step 1: Inputoutput model and scattering network expansion For a given Istructure I , construct its inputoutput description with the external environment under unied time scale: view internal degrees of freedom as nodes, external channels as waveguides or ports. Using standard systems theory methods, obtain scattering matrix family Snet(ω;λ) in frequency domain, where λ represents slowly varying internal parameters. Step 2: Identication of self-referential closed loops The dependence of internal updates on their own outputs manifests as feedback closed loops in frequency domain. Through Redheer star product, compress these closed loops into closed-loop scattering matrix S⟲(ω;λ) . View the entire network as a directed graph, perform strongly connected component decomposition, select minimal strongly connected components satisfying both memory and self-referentiality conditions, i.e., I-closed loops S(I) . Step 3: Consistency of delay spectrum and unied time scale By scale identity, calculate κ(ω) for S⟲(ω;λ) from scattering phase and time-delay operator, aligning with the unied time scale of Istructure. Require eective time density on I-closed loop to match time function on worldline in corresponding energy bandwidth. Step 4: Reversible steps and correspondence 9 A.5 Local Uniqueness up to Equivalence If another realization I′ ∗ is compatible with I∗ for the same records and unied time scale equivalence class, one can construct equivalence relation as follows: 1. Dene strictly monotone bijection f from two worldlines and time functions; 2. Using GNS representation and ∗ -isomorphism classication theory, construct ∗ -isomorphism Φ between Aint and A′int , being measure isomorphism on memory subsystem in spectral space; 3. Translate compatibility of observational data and internal updates into semigroup conjugacy condition for U and U′ , thereby satisfying Denition 7. Thus obtaining local uniqueness. Appendix B: Correspondence Between IStructures and Self-Referential Scattering Loops This appendix supplements key technical points for Theorems 2 and 3. B.1 From Internal Dynamics to Closed-Loop Scattering For a given Istructure I , construct linearized inputoutput model under unied time scale: view internal degrees of freedom as nodes, external channels as waveguides. In frequency domain, using appropriate degree of freedom choices and Laplace/Fourier transforms, convert time-domain update U(t2, t1) and environment maps Et into scattering matrix family Snet(ω;λ) . Self-referential update corresponds to feedback loops from output owing back to input. Use Redheer star product to compress feedback structure into closed-loop scattering matrix S⟲(ω;λ) , and verify it satises relative trace-class conditions. B.2 Modied Determinants and Spectral Shift On S⟲(ω;λ) , construct associated operator pair (H0, H) , using BirmanKren and related results to dene modied determinant detpS⟲ and spectral shift function ξp . Phase index map s(ω, λ) = det pS⟲(ω;λ) = exp−2πiξp(ω, λ) maps parameter space to unit circle. B.3 Z2 Holonomy and Minimal Strongly Connected Components Remove singularity set where s= 0 from parameter space, obtaining X◦ . Consider double cover dened by square-root determinant. Along closed path γ⊂X◦ , dene ν√S⟲(γ) = expiIγ 1 2is−1ds, giving Z2 index. Strongly connected component decomposition and spectral ow additivity show this index cannot be further decomposed on minimal strongly connected components, hence can use ν(S) to label topological type for each I-closed loop. 16 B.4 Alignment with NullModular Double Cover and BF Theory In NullModular double cover and Z2 BF theory, sector structure of local geometry and modular ow is represented by some cohomology class [K] on H2(Y, ∂Y ;Z2) . For physical systems satisfying local energy and entropy consistency, globally require [K] = 0 , i.e., no anomalous sectors topologically overall. The Z2 index of I-closed loop is conned to internal scattering subsystem, not disrupting overall geometry's cohomology class. This localglobal allocation allows self-referential fermionic ngerprint to exist within the subject interior without introducing new topological sectors at universe scale, thereby realizing localization of subjectivity topology. Through the above appendix constructions and proof outlines, one can see the one-to-one correspondence between Istructures and self-referential scattering closed loops, and stability of Z2 topological ngerprint, thereby supporting the unied mathematical denition of self in the main text. 17