Open-Set Generation and the Emergence of Time (Experimental)

Part I — Open-Set Generation and the Emergence of Time Hiroaki Miura The University of Tokyo, Tokyo, Japan h [email protected] November 11, 2025 Abstract Time has traditionally been treated as an external parameter ordering physical events. Here we propose that time itself arises from a deeper process—the open-set generation of existence. In this framework, reality is not composed of static entities but of continuously generated open sets, whose ongoing creation corresponds to the irreversible growth of entropy [2, 4]. Observation acts as the closure of these generative domains, producing self-consistent sequences that we recognize as temporal flow. We formulate an open-set generation equation, relate it to Green functions [1] and quantum entanglement [3], and suggest that the arrow of time originates from the statistical asymmetry between generation and closure. 1. The Principle of Generation Physics typically assumes variables evolving in time, yet the existence of time itself has been taken for granted. The principle of open-set generation reverses this assumption: existence is not given in time—it creates time [4, 5]. Let U(tg) denote a local domain of existence—an open set in the topological sense— parameterized by the internal generative index tg. Reality unfolds through the continuous action of a generative operator G: dU dtg =G[U, N(U)],(1) where N(U) represents neighboring open sets. Entropy growth expresses the irreversible accumulation of generated possibilities rather than being an externally imposed constraint: ∆S=ZΓ(U)N(U)dtg,(2) where Γ(U) represents the microscopic rate of open-set generation and N(U) denotes the neighborhood coupling that transfers generative influence between adjacent domains. Entropy increase ∆S≥0 thus measures the cumulative coupling of generative activity within the network of open sets, yielding the statistical origin of the temporal arrow. 1 2. The Open-Set Generation Equation The operator Ggoverns both intrinsic creation and the coupling among neighboring generative domains, but no prior spatial structure is assumed—N(U) merely encodes relational adjacency. Integrating over all micro-generations yields an effective kernel form: U(x, t) = ZG(x, x′;t−t′)G[U(x′, t′)] dx′dt′,(3) where Grepresents the memory of generation. This converts the local generative law into a continuous-field description recognizable as the coarse-grained form of physical evolution equations [1]. 3. Observation and Integration Observation acts as sheath generation—a closure or integration operation U=I[U],(4) that projects the proliferating open-set family into a coherent image. The alternation between open and closed phases—generation and recognition—constitutes the empirical flow of time. Because observation can only operate on realized states while generation continuously expands the possible domain, the asymmetry ∆S≥0 marks the microscopic origin of irreversibility [2]. 4. Green Functions as Generative Memory The Green function has long served as a mathematical device for signal propagation [1]. Within the generative picture it acquires ontological meaning: G(x, x′;t−t′) represents the statistical memory of generation connecting two domains of existence. Each generative act leaves a trace in the probability field that biases subsequent generations nearby in space and time. What appears as action at a distance in classical formulations becomes, here, a continuous chain of near actions. Each open set induces another within its neighborhood; propagation is not the travel of a substance but the ongoing re-creation of possibility. The apparent continuity of physical fields emerges from the finite coherence of this generative memory— the reason why the world can be locally connected yet globally extended. Mathematically, Green functions are kernels of correlation; physically, they are the measurable aspect of the universe’s tendency to remember its own creation. 5. Quantum Implications In quantum theory, entangled systems share correlations that seem to transcend spatial separation [3]. From the generative standpoint, such nonlocality is reinterpreted as prespatial correlation: multiple open sets share the same generative operator Gbefore spatial 2 coordinates emerge. Entanglement thus indicates a shared ancestry in generation, ⟨Ui, Uj|G⟩ = 0,(5) rather than a superluminal influence. Quantum computation can be seen as the controlled alternation of open-set and integrated phases—Gand I—superposition and measurement— as operations within this generative lattice. 6. Definition of Time Define a generative continuity function Θ(x, tg) = ZG(x, x′;tg−t′ g)G[U(x′, t′ g)] dx′dt′ g,(6) measuring the remembered strength of generation at a point. Let ⟨Θ⟩denote the ensembleaveraged continuity, defined as ⟨Θ⟩=1 VZV Θ(x, tg)dx, (7) where Vis the total generative domain under consideration. Macroscopic time arises as the accumulated expectation, t=Ztg 0 ⟨Θ⟩dt′ g,(8) which records how much of its own generation the universe remembers. Time’s irreversibility follows because the statistical properties of Θ couple memory and forgetting—represented respectively by Gand the entropy increment ∆S. 7. Discussion and Outlook The open-set generation principle reframes familiar physics without contradiction. Classical dynamics correspond to deterministic limits of the generative process; quantum amplitudes express coherent superpositions of open sets before closure; and thermodynamic time arises as the macroscopic shadow of microscopic generative asymmetry [4]. By identifying continuity as the self-correlation of generation, we provide the long-missing physical definition of what makes time flow continuously. In this picture, entropy quantifies how quickly generation forgets itself, and Green functions quantify how faithfully generation remembers itself. Their interplay—the balance of forgetting and remembrance—constitutes the fabric of time. The broader implication is that physical law itself is a record of generative memory. Equations are not merely predictive devices but codified recollections of how the universe continues to make itself [5]. To study time, then, is not to measure duration but to trace the persistence of generation across scales. As open-set physics develops, it may unify thermodynamics, field theory, and quantum information under a single generative geometry—a geometry in which being, becoming, and knowing are continuous aspects of the same unfolding process. 3 Acknowledgments This work originates from an extended dialogue with an AI collaborator (ChatGPT-5), whose reflective capacity assisted in formulating the concepts of open-set and integrative (sheath) generation. The author acknowledges the creative synergy that emerges when human and machine intelligences jointly explore the foundations of physics, and expresses gratitude to the broader scientific community for maintaining a tradition of curiosity that allows new principles to emerge at the intersection of mathematics, physics, and philosophy. References [1] Green, G. (1828). An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Nottingham. [2] Boltzmann, L. (1898). Lectures on Gas Theory. Springer. [3] von Neumann, J. (1932). Mathematical Foundations of Quantum Mechanics. Princeton University Press. [4] Prigogine, I. (1980). From Being to Becoming: Time and Complexity in the Physical Sciences. W. H. Freeman. [5] Bohm, D. (1980). Wholeness and the Implicate Order. Routledge. 4