Equivalence Characterization of Causal Structure and Potential Observer Category: Unified Time Scale, No-Local-Observer Limit, and Universe Ontology

Equivalence Characterization of Causal Structure and Potential Observer Category: Unied Time Scale, No-Local-Observer Limit, and Universe Ontology Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract Within the framework of unied time scale, boundary time geometry, and GLS universe objects, this paper provides a rigorously formalized answer to the question: Can causality be viewed as a product of observers? The core conclusions are: 1. Given a geometricdynamical universe object Ugeo = (X, ⪯, M, g, ˆ H, . . . ) , we can construct a category Obspot consisting of all potential observers, whose objects are timelike worldlines with memory systems, and whose morphisms are coarse-graining maps preserving causal and informational consistency. 2. Using the reachable memory structure of Obspot , we can reconstruct a partial order ⪯obs without explicitly referencing (X, ⪯) , representing event precedence relations that can be stably recorded in memory by some potential observer. 3. Under natural physical assumptions of locality, information reachability, and decoherence stability, we prove that ⪯obs=⪯ . Therefore, the causal structure of the universe and the potential observer category are mutually equivalent descriptions in a natural sense. Based on this, we distinguish between the potential observer category and the actual observer subset A ⊂ Obj(Obspot) . The so-called no-observer limit does not completely erase all observational structure, but merely sets A=∅ (no local observers), while the potential observer category and geometricscattering layer still exist. The global pure state of the universe ρglobal(t) and the unied time scale κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω) can be interpreted as the internal memory and time scale of the universe self-referential superobserver, thereby dissolving the ontological tension of whether a no-observer universe still has causality. The main theorem of this paper shows that under the GLS universe object axiom system, causal structure as memory reachability relation of potential observers is a rigorously provable equivalence, while actual observer networks are merely activated subfamilies within the potential observer category. The appendices provide formalized category constructions, denitions of reachable partial orders, and consistency proofs between unied time scale and decoherence quantum Darwinism. Keywords Causal structure; Potential observer category; Unied time scale; GLS universe objects; No-localobserver limit; Memory reachable partial order; Quantum Darwinism 1 1 Introduction Observers play a dual role in contemporary discussions of quantum gravity, quantum information, and cosmological ontology: On the one hand, general relativity and quantum eld theory, given M, gµν and local Hamiltonian ˆ H , seem capable of dening causal structure, time evolution, and scattering processes without any mention of observers; on the other hand, quantum measurement, decoherence, quantum Darwinism, and relational quantum mechanics emphasize that all physically accessible events and causality are always realized through some class of information carriers and memory systems (i.e., generalized observers). This paper attempts to provide a consistent mathematical answer to the following questions: 1. Can the causal partial order (X, ⪯) of the universe be equivalently understood as the aggregate structure of memory reachability relations of all potential observers? 2. In the no-observer limit (especially in the early universe), how do causal structure and unied time scale maintain their ontological existence in the absence of local observers? 3. Within the framework of unied time scale κ(ω) , boundary time geometry, and GLS categorical universe objects, how can global state objectivity, local state relationality be compatible with causalobserver equivalence? Existing discussions often remain at the interpretational level: e.g., whether the moon exists when no one observes it, whether causality depends on observers, etc. The goal of this paper is not to propose a new philosophical position, but to provide a formalizable, theoremizable answer within the already constructed framework of unied time scale and GLS universe objects, such that:  geometricscattering causal structure and  memory reachability of potential observer category mutually reconstruct each other mathematically, thereby realizing a rigorous version of causality as observer in the sense of categorical equivalence. The main thread of this paper is as follows: Section 2 reviews the core components of GLS universe objects and unied time scale; Section 3 denes the potential observer category Obspot ; Section 4 introduces the partial order ⪯obs induced by the memory structure of potential observers, and proves ⪯obs=⪯ under appropriate assumptions; Section 5 discusses the no-local-observer limit and the ontological status of the universe super-observer; Section 6 analyzes the compatibility of unied time scale, decoherence, and quantum Darwinism within this framework. Appendices provide formalized proof details and several technical lemmas. 2 GLS Universe Objects and GeometricScattering Causal Structure 2.1 GeometricDynamical Universe Object We adopt a class of abstract GLS universe objects, whose geometricdynamical layer provides the following data: 2  Event set X , understood as localizable events on spacetime manifold M with some discrete or continuous indexing;  Causal partial order ⪯⊂ X×X , satisfying reexivity, antisymmetry, and transitivity, compatible with the light cone structure of M, gµν ;  Metric geometry (M, g) , or its discrete substitute in QCA universe, such as causal networks with bounded degree, lattices, etc.;  Local dynamics: either Hamiltonian ˆ H with corresponding unitary evolution U(t) = exp(−iˆ Ht) , or discrete-time quantum cellular automaton update operator U , whose local support respects causal structure. Denition 1 (GeometricDynamical GLS Universe Object) . A geometricdynamical GLS universe object is Ugeo = (X, ⪯, M, g, ˆ H, . . . ), where (X, ⪯) represents the causal partial order, (M, g) is the continuous geometric background (or QCA substitute), ˆ H or U is the local dynamics, satisfying micro-causality and energy conditions among standard assumptions. At this level, causal structure is viewed as a geometricdynamical ontological structure, independent of the existence of actual observers. 2.2 Unied Time Scale and Scattering Causality The unied time scale is provided by scattering theory. Consider a class of scattering systems satisfying trace-class perturbation and wave operator completeness conditions. The total phase φ(ω) of the scattering matrix S(ω) , the spectral shift function density ρrel(ω) , and the Wigner Smith time-delay matrix Q(ω) = −iS(ω)†∂ωS(ω) satisfy the unied scale identity: κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). Here κ(ω) can be interpreted as the mother time scale density dened in the frequency domain, whose integral gives eective arrival time, delay accumulation, etc. The relationship between unied time scale and causal structure can be roughly understood as:  The geometricdynamical layer determines which events can inuence each other (causal partial order);  The scatteringscale layer provides the temporal weight and resolution of these causal relations in the frequency/energy domain. In this paper, we assume that the unied time scale is constructed completely within the GLS framework; the focus is on how to embed the observer layer on this foundation and formally realize causalobserver equivalence. 3 3 Construction of Potential Observer Category This section denes the concept of potential observer, making it depend only on the geometric dynamical universe object Ugeo , independent of whether actual observers exist in any particular universe history. 3.1 Worldline and Memory System of Potential Observer Intuitively, an observer requires at least: 1. A worldline respecting the causal partial order (timelike trajectory); 2. An internal memory system evolving along the worldline; 3. Ability to interact with local observable algebras and update memory. Denition 2 (Potential Observer Object) . Given a GLS universe object Ugeo , the data of a potential observer O includes: 1. A causal chain LO⊂X , i.e., for any x, y ∈LO , either x⪯y or y⪯x ; 2. A family of memory Hilbert spaces and states (HO, µ(τ)) evolving with parameter τ ; 3. For each x∈LO , there exists a local observable algebra A(x)⊂ B(Hx) and CPTP channel Φx:B(Hx)⊗ B(HO)→ B(HO), describing the process of reading information from local system and writing to memory. We call O a potential observer when its (HO, µ(τ),Φx) is compatible with the local dynamics ˆ H or QCA rules of Ugeo , i.e., does not violate micro-causality and energy constraints. Denition 3 (Potential Observer Category) . Let Obspot be the category where:  Objects are all potential observers O ;  Morphisms f:O → O′ are maps preserving worldline causal ordering and memory information reachability, including reparametrization, memory coarse-graining, and internal encoding transformations, such that f does not introduce superluminal information ow. Intuitively, Obspot describes all observer worldlines and memory structures that are physically possible under a given geometricdynamical universe object, without requiring them to be actually activated in any particular universe history. 3.2 Actual Observer Subset In a concrete universe history, the observers that truly emerge occupy only a small fraction of the potential observer set. Denition 4 (Actual Observer Set) . The actual observer set A is dened as a subset of Obspot : A ⊂ Obj(Obspot), representing observers whose memory systems are actually activated and participate in information storage in a given universe evolution history. 4 Thus, the no-local-observer limit should be more precisely stated as: |A| = 0 rather than Obspot being empty. The potential observer category still exists as determined by Ugeo . 4 Equivalence of Causal Partial Order and Observer Memory Reachable Partial Order This section constructs a partial order ⪯obs determined solely by the potential observer category, and proves under reasonable assumptions that it coincides with the geometricdynamical causal partial order ⪯ . 4.1 Dening Reachable Partial Order from Potential Observers Denition 5 (Memory Reachability Relation) . For any x, y ∈X , dene the relation x⪯obs y as: There exists some potential observer O ∈ Obspot , and parameters τx< τy , such that: 1. x, y ∈LO and correspond to worldline points at times τx, τy ; 2. In the memory state µ(τy) at time τy , information about event x can still be recovered through some observable means (possibly coarse-grained). Denoted as: x⪯obs y⇐⇒ ∃O, τx< τy:x, y ∈LO,Infox,→µ(τy). Here Infox,→µ(τy) indicates that there exists some observable algebra element, POVM, or post-processing procedure that can recover statistical information about x from µ(τy) . Proposition 6. Under the above denition, ⪯obs is a partial order relation on X . Proof. Reexivity is obtained by taking x=y without evolving memory; antisymmetry and transitivity depend on the causality of observer worldlines and monotonicity of memory updates; see Appendix A.1 for details. 4.2 Geometric Causality Implies Memory Reachable Partial Order Theorem 7 (Geometric Causality Implies Memory Reachability) . If x⪯y (geometricdynamical causal partial order), then under locality and observability assumptions, we must have x⪯obs y . Proof (Sketch). From x⪯y and the micro-causality of Ugeo , there exists a causal curve (timelike or null) extending from x to y . Along this curve, construct a potential observer O whose worldline LO contains x, y , and at time x interacts with the local observable algebra A(x) , writing some information about x into memory µ(τx) . Since local dynamics and energy conditions guarantee that information is not completely annihilated in nite time, there exists τy> τx such that µ(τy) still retains recoverable Infox . Thus x⪯obs y . Formalized proof in Appendix A.2. This theorem shows that every causal relation given by the geometricdynamical layer can be realized by the memory trajectory of some potential observer, hence ⪯⊆⪯obs . 5 4.3 Memory Reachable Partial Order Implies Geometric Causality The more constraining half is the converse: Theorem 8 (Memory Reachability Implies Geometric Causality) . In GLS universe objects assuming no superluminal information ow, if x⪯obs y , then we must have x⪯y . Proof (Sketch). Assume x⪯ y and y⪯ x , i.e., x, y are spacelike separated or incomparable. If there exists a potential observer O whose memory µ(τy) at time τy still retains recoverable Infox , this implies an information ow channel from x to y passing through the observer's internal degrees of freedom. But under GLS axioms, the observer is also just a physical subsystem of the universe, whose internal propagation obeys the same causal partial order. Therefore, memory reachability from x to y implies the existence of some timelike curve from x leading to y , contradicting the spacelike separation assumption. Formally, the observer can be viewed as a local system embedded in M , whose internal propagation cone is contained within the background light cone. If the memory reachability structure can transmit information between x, y , then necessarily x⪯y . See Appendix A.3 for details. From Theorems 4.3 and 4.4, we immediately obtain: Corollary 9. Under the locality and micro-causality assumptions of GLS universe objects, ⪯obs=⪯. This shows that the geometricdynamical causal partial order of the universe is completely equivalent to the partial order of memory reachability relations of all potential observers. In other words, causal structure and potential observer category are mutually equivalent presentations of the same universal terminal object in a natural sense. We can therefore understand causal structure as memory structure of potential observer networks without loss of generality, and vice versa. This is the core mathematical result of this paper. 5 No-Local-Observer Limit and Universe Super-Observer 5.1 Reinterpretation of No-Local-Observer Limit In previous discussions of no-observer universe, we consider the limit |A| → 0 . Combined with the equivalence theorem of the previous section, we can provide a more precise formulation. Denition 10 (No-Local-Observer Limit) . In GLS universe objects, the no-observer limit should be strictly understood as the no-local-actual-observer limit: |A| = 0,Obspot =∅. At this point, the geometricdynamical causal partial order (X, ⪯) and potential observer category Obspot still exist, the memory reachable partial order ⪯obs and ⪯ still satisfy the equivalence theorem; only no subsystem is actually activated as a local observer. Physically, this corresponds to the early universe or structureless universe: geometry, elds, scattering, and unied time scale already exist, causal structure exists as an ontological object, but no complex subsystem has yet formed a stable memory carrier. At this point, causality as potential observer network still holds, only all observers are in the potential state. 6 5.2 Universe as a Whole as Super-Observer In the GLS framework, the global pure state of the universe ρglobal(t) can be viewed as a selfreferential super-observer memory. Denition 11 (Universe Super-Observer) . Dene a special potential observer Ouniv : 1. LOuniv =X , i.e., its worldline traverses all events in an abstract sense; 2. The memory Hilbert space is the global Hilbert space Hglobal , with memory state ρglobal(t) ; 3. Memory update is given by unitary evolution U(t) = exp(−iˆ Ht) , i.e., ρglobal(t) = U(t)ρglobal(0)U(t)†. Although this super-observer is dicult to concretize as any local physical entity, ontologically it can be viewed as an EBOC-style eternal block observer: its memory is the global pure state of the universe, recording the statistical structure of all possible events in the most detailed manner. Proposition 12. In the no-local-observer limit |A| = 0 , the universe still has a unique superobserver Ouniv , whose memory state ρglobal(t) and corresponding causalscattering structure completely determine the causal partial order ⪯ and potential observer category Obspot . Therefore, no-observer universe in the strict GLS sense should be understood as no-localobserver universe, not no-observer-whatsoever universe. The universe as a whole can always be viewed as its own super-observer, whose self-referential memory structure together with the unied time scale denes the universe ontology. 6 Compatibility of Unied Time Scale, Decoherence, and Quantum Darwinism This section explains how unied time scale and decoherencequantum Darwinism naturally embed into the causalobserver equivalence framework, resolving the controversy of whether wavefunction collapses due to observers. 6.1 Decoherence as Information Diusion in Potential Observer Network Consider the standard decoherence model of system S and environment E . Initial pure state |Ψ(0)⟩=|ψS⟩⊗|0E⟩ evolves under interaction Hamiltonian to entangled state |Ψ(t)⟩=X i ci(t)|iS⟩⊗|ϕE i(t)⟩. If |ϕE i(t)⟩ are approximately orthogonal, the reduced system state ρS(t)≈X i |ci(t)|2|iS⟩ ⟨iS| appears as a diagonal classical mixture. This process can be viewed as: a large number of potential observers in the environment (such as local degrees of freedom of environment subblocks 7 Ek ) redundantly record pointer state information about S in their respective memory degrees of freedom. In the framework of this paper, this corresponds to:  There exist many objects in the potential observer category Obspot whose worldlines pass through events where S occurs, storing redundant information about |iS⟩ in memory states;  Pointer states are the encodings most stable and redundant in the memories of these potential observers. Quantum Darwinism further emphasizes: classical objectivity arises from redundant copying of information in the environment; in our language, this is equivalent to: certain event sets leave consistent traces in the memories of many objects in Obspot , thereby forming causal structure with broad consensus on ⪯obs . 6.2 Unied Time Scale as Time Parameter of Super-Observer The unied time scale κ(ω) in this framework is naturally interpreted as the frequency-domain time coordinate of the super-observer.  For the super-observer Ouniv , κ(ω) provides a consistent scale for scattering phase derivative, spectral shift density, and WignerSmith delay trace, serving as the mother scale for the universe's overall frequency-domain causal structure.  For any local potential observer O , its local temporal experience can be viewed as some sampling and coarse-graining of κ(ω) , e.g., measuring the local delay spectrum through local scattering processes to dene its own time. The causalobserver equivalence theorem guarantees that whether from the geometricscattering side (via κ(ω) and ⪯ ) or from the observer network side (via potential observer memory and ⪯obs ), the resulting time arrow and causal ordering are consistent. 6.3 GLS Version of Wavefunction Ontology In this framework, the wavefunction (or more precisely, the global density operator) plays a dual role: 1. As the memory state ρglobal of the super-observer Ouniv , it is part of the universe's ontological structure, independent of any specic local observer; 2. As the reduced state ρα= trCα(ρglobal) of local observer Oα , it is a relational object relative to its causal fragment Cα . Decoherence and quantum Darwinism explain: the stability of local reduced states in the potential observer network determines the emergence of classical reality, not the introduction of consciousness or specic types of observers. The causalobserver equivalence further shows that these stable structures can be completely rewritten in terms of potential observer memory reachability relations, equivalent to the geometricscattering causal partial order. 8 7 Discussion and Prospects The causalpotential observer category equivalence construction presented in this paper provides a self-consistent ontological picture for the relationships among unied time scale, GLS universe objects, and no-local-observer limit:  Geometricscattering side: describing the causal and temporal structure of the universe via (X, ⪯, M, g, ˆ H) and κ(ω) ;  Observer side: describing all feasible observational structures via potential observer category Obspot and memory reachable partial order ⪯obs ;  Equivalence theorem: ⪯obs=⪯ makes causality and observer network two presentations of the same universal terminal object, naturally reinterpreting no-observer universe discussions as no-local-observer but with super-observer and potential observer universe. Future work directions include: in specic QCA universe models and THE-MATRIX Universe, explicitly constructing Obspot , and verifying whether causalobserver equivalence still holds under nite size and nite information conditions, and how it further couples with black hole information paradox, cosmological constant problem, etc. Appendix A: Formalized Properties and Proofs of Memory Reachable Partial Order A.1 Partial Order Properties of Memory Reachability Relation In Denition 4.1, the memory reachability relation ⪯obs must satisfy the three partial order axioms. Reexivity. For any x∈X , construct a degenerate potential observer Ox whose worldline contains only event x , with memory system reading its own state or external metric information at x and immediately writing back. Then x⪯obs x . Antisymmetry. If x⪯obs y and y⪯obs x , then there exist potential observers O1,O2 and parameters τx< τy , τ′ y< τ′ x , such that O1 's memory at τy contains information about x , and O2 's memory at τ′ x contains information about y . This means information ows exist from x to y and from y to x . Under GLS axioms' no-closed-causal-curve and no-superluminal-propagation conditions, this can only occur in the degenerate case x=y (otherwise constructing closed timelike curves or superluminal signals). Thus if x⪯obs y, y ⪯obs x then x=y . Transitivity. If x⪯obs y and y⪯obs z , then there exist potential observers O1 and O2 whose memories at appropriate times contain Infox and Infoy respectively. Construct a new potential observer O3 traveling along a composite causal path containing x, y, z , inheriting memory contents from O1,O2 along the way (this is allowed, as Obspot permits coarse-graining and merging maps). Then O3 's memory when passing through z can contain Infox , thus x⪯obs z . Therefore ⪯obs is a partial order. A.2 Detailed Proof: Geometric Causality Implies Memory Reachability Given x⪯y , this means there exists a timelike or null curve γ: [0,1] →M satisfying γ(0) = x, γ(1) = y , with γ always lying within the future light cone of x and past light cone of y . 9