Continuous Time, Discrete Observations, and Entropy: A Minimal Model in Which Time Emerges from Observation

Continuous Time, Discrete Observations, and Entropy: A Minimal Model in Which Time Emerges from Observation Hiroaki Miura The University of Tokyo, Graduate School of Science h [email protected] December 3, 2025 Abstract This study presents a minimal model demonstrating how the structure of time can emerge solely from observational operations, without assuming geometry, metrics, or physical laws. A countably infinite sequence of selections of generative bases determines observation and defines an irreversible arrow of internal time. Because the set of all such selection histories has the cardinality of the real numbers, the completion points of observations lie on the real line. Thus, a discrete set of observation moments becomes embedded within continuous time, yielding a reversible arrow of external time. Furthermore, the finiteness of entropy — defined as the finite change in the cardinality of generative bases retained in observation — is required to ensure the continuity and consistency of the observable world. In this framework, the continuity and discreteness of time, as well as its reversibility and irreversibility, emerge naturally from observational selection operations. 1 Introduction In our observed world, the sun rises in the morning and sets in the evening. We can record the times of sunrise and sunset and attribute them to the Earth’s rotation, revolution, and axial tilt. Yet the very possibility of defining time so naturally invites deeper reconsideration. Many attempts have been made in various fields, such as quantum gravity and information thermodynamics, to derive temporal structure from physical foundations (Connes & Rovelli, 1994; Isham & Butterfield, 1999; Rovelli, 2009; Huggett & W¨uthrich, 2012; Oriti, 2021). However, these studies all assume specific physical premises, such as quantum states, geometry, or statistical entropy. In contrast, this study does not adopt such premises: it formally investigates how the seemingly contradictory properties of time — continuity and discreteness, as well as reversibility and irreversibility — emerge solely from observational operations. To this end, we consider an abstract world with a latent state space X∞, which possesses an uncountably infinite structure. Although X∞itself is not directly accessible, observation can reach it only in a limited, restricted manner. We assume that what becomes observable constitutes a countably infinite structure extracted from this latent space. Observation is modeled as a sequence of countably infinite selection operations. The admissible domain for these selections is given by B0⊂ P(X∞), a family of countably infinite subsets of X∞that may be selected during observation. Each element U∈B0is a countably infinite set. 1 An observation proceeds by selecting one element of B0at each step, forming a countably infinite sequence U1, U2, U3,.... Only when this infinite sequence is completed does the observation induced by the fluctuation C become definite; at any finite stage, nothing definite has yet been observed. In the above setting, we have assumed that the selection operations under observation can be described as an ordered, countably infinite sequence. Thus, an ordered structure resembling a “flow of time” is introduced into X∞. This order is the minimal structure required to represent observation as a countably infinite sequence of selection operations, and does not presuppose any notion of physical time. We do not investigate the latent ordering relation itself. Rather, our interest lies in how temporal structure in the observable world is generated: first, by each completed observation — defined as a countably infinite sequence — and then by the discrete succession of such completions. 2 Internal Time and Irreversibility To examine how temporal structure emerges, we first clarify the components of observation: the latent space X∞, the fluctuation Cdefined on it, the generation operator G, and the sequence of sheath sets (Σn)n∈N. Their roles are as follows. X∞is an uncountably infinite latent reservoir. Rather than assuming any topological structure on X∞, we introduce a latent family of candidate sets B∞⊂ P(X∞). Each U∈B∞is a countably infinite subset of X∞that can potentially be selected for observation. We refer to such Uas a generative basis. Cis an abstract fluctuation over X∞that induces the emergence of observation by determining which generative bases may be selected. Consequently, the generative bases available to observation constitute a countable subset B0⊆B∞with cardinality ℵ0. No intrinsic dynamics of Care assumed in this study; it serves solely as a criterion for observational admissibility. At each stage of the selection operation, the operator Gselects one generative basis from B0. The sheath set Σn(n∈N) records the bases selected up to stage n, with Σ0=∅. To formalize the admissibility condition for generative bases, we introduce the finite union family generated by Σn: O(Σn) = ([ k∈F Uk F⊆ {1,2, . . . , n},0<|F|<∞). The generation operator Gselects a new generative basis that cannot be expressed as a finite union of previously selected ones. Accordingly, the update rules are: Un=G(C, Σn−1), Un/∈ O(Σn−1), Σn= Σn−1∪ {Un}. In other words, the new generative basis Un, which cannot be expressed as any finite union of previously selected bases, is extracted from B0and appended to Σn−1to form Σn. If no admissible generative basis exists, we set Un=∅,Σn= Σn−1. Even in this case, the selection step is still regarded as performed, and the process proceeds with n7→ n+ 1. 2 An observation is defined to be complete when a countably infinite number of generative bases has been selected. At any finite stage Σn, the observation itself remains indeterminate; it becomes definite only when Σ∞=[ n∈N Σn is formed. Because each selection, once made, is irreversibly recorded into the sheath, the number of selected generative bases increases without bound. If no admissible new basis exists, we allow Un= ∅, indicating a stagnation of observable expansion while the observation process itself continues. Thus, the observation process proceeds unidirectionally, n7→ n+ 1, with no possibility of reversal. This structural irreversibility endows the process with a strictly non-decreasing temporal order. In this sense, the internal “arrow of time” emerges not from physical dynamics but from the irreversible accumulation of observational steps. 3 Emergent Continuity and External Time 3.1 Continuity of Time Through the countably infinite selection operations inherent in observation, a non-redundant sequence of generative bases {Ui}i∈Nis constructed, whose completion defines the sheath Σ∞. This sequence represents the full history of the extraction of generative bases from the countable subset B0⊆B∞⊆ P(X∞) within the latent state space X∞. We denote the history associated with the observation completed at the τ-th iteration by Σ∞(τ). According to Cantor’s theorem (Cantor, 1891), the set B∞has the cardinality of the continuum (|B∞|= 2ℵ0=|R|), whereas the set B0restricted by Chas countable cardinality (|B0|=ℵ0). Let H=BN 0denote the set of all possible histories of countably infinite selections on B0. Then its cardinality is |H|=|B0|ℵ0= (ℵ0)ℵ0= 2ℵ0=|R|. Thus, while each selection operation involves only countably many choices, their completion — as the limit of a countably infinite sequence — achieves the full cardinality of the continuum. In this sense, the continuity of time does not reside in any single selection step but in the limit structure of their countably infinite histories. Since the set Hhas the same cardinality as the set of real numbers, there exists a surjective function f:H→R. For an observation history Σ∞(τ)∈H, we define its associated moment by t(τ)=f(Σ∞(τ)) ∈R. We require t(τ) to be strictly increasing in accordance with the order of completed observations: τ < τ′⇒t(τ)< t(τ′). We do not specify the explicit form of the map f:H→R. Its existence follows solely from cardinality arguments, and the monotonicity condition above can always be imposed by choosing an order-preserving enumeration of the completed observations. Thus, {t(τ)}τ∈Nconstitutes a strictly increasing sequence of moments on a continuous real time axis, generating the external “arrow of time” among completed observations. Importantly, this construction requires only an ordering structure — no physical laws are presupposed. The continuity of time is not an assumption; rather, it emerges from the fact that the set of all countably infinite selection histories has the cardinality of the continuum. As τincreases, a time 3 axis with continuum cardinality naturally arises in the observable world, where each completed observation is assigned a definite moment. 3.2 Discreteness of Observations During the construction of Σ1,Σ2, . . ., no observation has yet occurred. The set of generative bases remains incomplete at any finite stage, and therefore no definite observed moment can be assigned. A definite observational result arises only once the countably infinite selection operations are completed and Σ∞is formed. At that moment, the completed observation is assigned a single external moment t(τ), where τindexes the number of completed observations. Therefore, the set of observation moments is given by Tobs ={t(τ)|τ∈N}. Each t(τ) is an isolated point on the real line. Yet behind each such point lies an internal process of continuum cardinality, namely the countably infinite selection operations required to complete an observation. In this sense, the time axis is a continuum on which observational results are discretely inscribed. 4 Entropy In the observations considered here, the limit of the generative basis sequence {Ui}i∈Nconstitutes the sheath set Σ∞(τ) corresponding to the completed τ-th observation. If the fluctuation Cdiffers between the τ-th and (τ+ 1)-th iterations, then a discrepancy may result between Σ∞(τ) and Σ∞(τ+ 1). We therefore introduce an indicator to quantify this difference. The family of unions generated by the sheath set Σ∞(τ) for the τ-th observation is defined as O(Σ∞(τ)) = ([ k∈F Uk F⊆N,0<|F| ≤ ℵ0). More explicitly, it consists of every union of finitely or countably many generative bases drawn from Σ∞(τ). Using this, the generative bases newly selected in the (τ+ 1)-th observation are given by Σ+= Σ∞(τ+ 1) \ O(Σ∞(τ)). Conversely, the generative bases that are not retained in the (τ+ 1)-th observation are represented as Σ−= Σ∞(τ)\ O(Σ∞(τ+ 1)). We refer to Σ+as the generative action and to Σ−as the sheath action, corresponding to the forgetting of unused generative bases. The entropy Sis defined to record the net change in the cardinality of generative bases retained in the observable world: S(t(0)) = 0, S(t(τ+ 1)) = S(t(τ)) + |Σ+|−|Σ−|. Entropy increases when more generative bases are retained to describe the observable world and decreases when fewer such bases remain necessary. 4 For the observable world to remain consistent and continuous, both the generative and sheath actions must remain finite, i.e., |Σ+|<∞and |Σ−|<∞. This finiteness condition must hold at every observational step. If infinitely many new generative bases must be selected and infinitely many previously chosen bases discarded, then observations lose their connection to past states, and the observable world effectively “tears apart.” Therefore, in order to maintain a consistent and continuous history, the entropy change |∆S|=|S(t(τ+ 1)) −S(t(τ))| must satisfy |∆S|<∞for every observational step. 5 Observable and Dark Sectors The τ-th completed observation determines the sheath set Σ∞(τ). At this point, the observable world (observable sector) is defined as the union of all generative bases recorded in Σ∞(τ): Uall(τ) = [ U∈Σ∞(τ) U. In contrast, the unobservable world (dark sector) is defined as the complement within the latent state space: Ωdark(τ)=X∞\Uall(τ). The temporal order that emerges from the countably infinite selection of generative bases is realized only within the observable sector Uall(τ). Conversely, no temporal ordering emerges in the dark sector Ωdark(τ). However, changes in the latent fluctuation Cmay affect both the observable and dark sectors. If Cchanges due to an action in the observable sector, a corresponding effect may also arise in the dark sector. But in that case, because no temporal ordering exists there, such changes do not appear as temporal evolution but remain unrecorded in time. Thus, the dark sector may change its structure despite lacking internal temporal order. Yet changes in the latent state space appear as temporal observations only within the observable sector, which is generated by countably infinite selection operations. 6 Conclusion This study formally examines how time emerges solely from the structure of observation, without assuming geometry, metrics, or physical laws. Entropy is introduced only after time emerges, to record changes in the cardinality of generative bases retained in the observable world. (i) The latent state space X∞contains an uncountably infinite family of generative bases B∞. Each generative basis U∈B∞has countable cardinality. (ii) The fluctuation Con the latent state space X∞restricts B∞to a countable subset B0, so |B0|=ℵ0. (iii) An observation consists of a countably infinite sequence of selection operations on the generative bases. At each step, the generation operator Gselects a new, non-redundant generative basis Un/∈ O(Σn−1) and the sheath set is updated by Σn= Σn−1∪{Un}.The τ-th observation completes when the limit Σ∞(τ) is formed, defining an irreversible internal “arrow of time.” 5 (iv) Each completed observation Σ∞(τ) belongs to the history space H=BN 0, whose cardinality equals that of the continuum, |H|= 2ℵ0=|R|. Based on the order of completed observations, a monotonically increasing real-valued time t(τ) is assigned to Σ∞(τ), giving rise to the external ”arrow of time.” (v) The observation moments form a discrete set Tobs ⊂R, yet they are situated on a time axis of real cardinality. (vi) The entropy Srecords the change in cardinality of the generative basis sequence. For the observable world to remain continuously maintained, it is necessary that the entropy change satisfies |∆S|<∞at each observational step. (vii) Observation separates the latent state space X∞into the observable sector Uall(τ) and the dark sector Ωdark(τ). The arrow of time exists only in the observable sector; no temporal structure is defined in the dark sector. Changes in the fluctuation Cmay influence not only the observable sector but also the dark sector, even though such changes cannot be temporally ordered there. Within the observable sector, internal time is irreversible and lacks time-reversal symmetry. Conversely, the ordering structure of the external time t(τ) assigned to completed observations retains arbitrariness, being unconstrained by physical laws, and can in principle remain invariant under the time-reversal transformation t7→ −t. Thus, the temporal structure emerging in this formal world exhibits a dual-layered arrow of time: time-reversal symmetry is broken internally, while it may still hold externally. 7 Discussion This study has formally demonstrated what kind of temporal structure emerges from specific observational operations, without assuming any spatial structures or physical laws. We briefly address the arbitrariness inherent in this formal setting, and also consider how such a framework may connect to numerical simulations. 7.1 Arbitrariness in the Selection Process The continuity of time is achieved by establishing the sheath set Σ∞via countably infinite selections from the countably infinite set B0. Yet arbitrariness persists: both the specification of B0and the selection process remain unconstrained in this abstract framework. First, restricting the cardinality of B0to countable infinity is the minimal condition that reconciles observability with the continuity of time. If B0were uncountable, the selection process could never be completed, and observation would be undefined. Conversely, if B0were finite, the cardinality of the history space would fail to reach the continuum, and real time would not emerge. Thus, the minimal cardinality of countable infinity elevates the history space H=BN 0to continuum cardinality, thereby generating real time in the observable world. Next, there remains flexibility in configuring the selection operation. Mathematically, any sequential selection of generative bases is admissible, and many possibilities exist. However, most such choices implicitly rely on spatial structures or physical fields. To avoid these assumptions, we have adopted an abstract setting in this study. Even so, the selection operation still retains a degree of modeling freedom. 6 Whether internal time in the real world exists and progresses discretely remains unclear. Instead, the emergence (generation) of structure from the latent state space into the observable world and its reinclusion (sheathing) from the observable world back into the latent state space may be regarded as a continuous process, with both fluctuating in dynamic equilibrium. Regardless of the specific implementation, continuous time is realized as long as the countably infinite selection history attains continuum cardinality. Here, we demonstrated the emergence of temporal continuity within a minimal discrete model. Constructing smoother models of internal time — such as those based on dynamic equilibrium between generation and sheathing — remains a challenge for future work. 7.2 Axiomatic Limit Continuous time is not assumed in this framework; rather, it emerges from two axiomatic leaps that underlie the observational process. We examine each of them below. 7.2.1 Restriction from the Latent State Space to the Observable World We assume that the latent state space admits an uncountably infinite family of generative bases, B∞⊂ P(X∞),|B∞|= 2ℵ0. We restrict this to a countable subset that ensures the observability and definability of the selection operation: B0⊂B∞,|B0|=ℵ0. Extracting a countable subset from an uncountable family requires an axiom-of-choice–type operation that cannot be avoided at this level. However, the emergence of continuous time in this study does not require X∞itself to be uncountably infinite. It may instead be countably infinite. In this case, no axiom-of-choice–type operation is needed to define B0, since both B∞and B0are already assumed to have countable cardinality in this setting. If the set B∞is countably infinite and its subset B0is also countably infinite, then the set H=BN 0of all histories of countably infinite selections on B0has the cardinality of the continuum. Therefore, even if X∞is countably infinite, each completed observation can still be assigned a moment on a continuous time axis. 7.2.2 Axiomatic Introduction of the Observation Limit The sequence of sheath sets constructed through successive applications of the generative operator G, Σ0⊂Σ1⊂···⊂Σn⊂ · · · , remains observationally indeterminate at every finite stage. An observation is completed only when an axiomatic operation identifies the limit Σ∞=[ n∈N Σn as a single, well-defined observational event. Introducing this limit allows external time to be assigned exclusively to completed observations: Tobs ={t(τ)|τ∈N}⊂R. 7 7.3 Relation to Numerical Simulations The temporal structure demonstrated in this study is isomorphic to the structure of time generally employed in physical numerical simulations. In numerical models, continuous time is assumed first, and then discretized by introducing finite-width time steps ∆t. This ∆tcorresponds to the intervals between discrete observation times in the external time of our framework. Meanwhile, the adjustment processes intrinsic to implicit time integration methods may be interpreted as operating within the internal time. However, whereas continuity and discreteness are assumed as given in numerical simulations, in this study they emerge from observational selection operations. This correspondence implies that numerical time-stepping schemes may already be incorporating — albeit implicitly — the more fundamental temporal structure revealed in this framework. The finite entropy change condition (|∆S|<∞) obtained in this study is structurally analogous to the stability conditions imposed on time integration in numerical simulations. The requirement that “the observable world remains continuously well-defined” and the requirement that “discrete time integration does not break down” may therefore reflect isomorphic structural constraints. Clarifying this correspondence remains a challenge for future work. Acknowledgments This work was developed through ongoing dialogue with ChatGPT-5, and its structure and expression were shaped in collaboration with AI. What is presented here is the result of a small experimental attempt to create a conceptual world together with AI. I am neither a mathematician nor a theoretical physicist, and this work may well contain serious misunderstandings. I express my gratitude to Zenodo for enabling the public dissemination of such a personal and exploratory experiment. References Cantor, G. (1891). ¨ Uber eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutschen Mathematiker-Vereinigung, 1, 75–78. Connes, A., & Rovelli, C. (1994). Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories. Classical and Quantum Gravity, 11(12), 2899– 2918. Huggett, N., & W¨uthrich, C. (2012). Emergent spacetime and empirical (in)coherence. 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