Recursive Structural Reality: A Philosophical and Mathematical Framework for a Cyclic Universe

Abstract

This thesis develops Recursive Structural Reality (RSR) as a unified philosophical and mathematical framework for investigating the fundamental organization of reality, the emergence of physical structures, and the long-term evolution of the universe. Its central proposition, Identity = Stable Recursion, suggests that the persistence and identity of a system can be understood through the recursive processes that maintain and transform its organization over time. Rather than treating existence as a collection of isolated objects, RSR approaches reality as a relational and dynamically organized process in which structures emerge, interact, transform, and persist through patterns of recursive continuity. The conceptual foundation of RSR is expressed through the sequence: Existence → Difference → Relation → Structure → Information → Transformation → Recursion → Stability → Persistence → Identity. This sequence provides an organizing framework for examining how distinguishable entities arise from relations, how relations generate organized structures, and how recursive processes may sustain recognizable identities despite continuous change. The thesis develops this framework across philosophy, mathematics, information theory, geometry, gravitation, and cosmology, distinguishing the conceptual principles of RSR from the mathematical and physical models proposed to investigate them. The mathematical programme begins with relational networks and graph structures, recursive state spaces, dynamical operators, and criteria for stability and instability. It then investigates how large-scale geometric properties might emerge from underlying relational organization and how recursive dynamics could be connected to effective spacetime, gravitational behaviour, and cosmological evolution. Standard mathematical tools—including graph Laplacians, dynamical systems, eigenvalue analysis, continuum limits, and cosmological field equations—provide a basis for formulating and evaluating candidate models. A central objective is to establish a mathematically consistent connection between microscopic relational dynamics and the macroscopic geometry and evolution of the universe. The cosmological component explores the possibility that the universe's evolution may involve phases of expansion, structural instability, contraction, and transition into an extreme physical regime from which a renewed expanding phase could emerge. Within RSR, this possibility is investigated through the concept of recursive cosmic renewal. The proposed renewal process does not require the successor universe to reproduce the preceding universe's complete physical state. Instead, it allows for the possibility that selected structural relations, constraints, or organizational principles persist across a transition even when the effective physical state changes. This distinction between recursive continuity and exact state identity provides a conceptual basis for exploring a universe that may renew without repeating exactly. The thesis also examines the relationship between cosmic identity, information continuity, entropy, gravitational collapse, and the mathematical conditions required for a viable transition between cosmological phases. It considers how a proposed renewal mechanism would need to satisfy appropriate dynamical constraints, preserve mathematical consistency, account for information and entropy, and produce a stable subsequent phase of evolution. Particular attention is given to the distinction between black-hole collapse and a hypothetical universe-wide transition, as well as to the challenge of deriving effective spacetime and gravitational dynamics from a more fundamental relational description. RSR is presented as a developing framework whose philosophical principles, mathematical architecture, and cosmological implications must be connected through explicit derivations and testable models. The thesis therefore distinguishes established mathematical results from proposed RSR constructions and identifies the principal research challenges: establishing a rigorous continuum limit, deriving effective gravitational dynamics, determining the conditions for cosmological turnaround and renewal, analysing the stability of successor phases, and identifying observational consequences that could distinguish the framework from existing cosmological models. By integrating relational ontology, recursive organization, mathematical stability, and cosmological renewal, this thesis aims to establish a systematic foundation for investigating whether persistence, identity, and large-scale cosmic evolution can be understood through a common structural principle. Its broader objective is to develop Recursive Structural Reality into a coherent research programme connecting philosophical inquiry with mathematical modelling and physical investigation, while exploring the possibility that the universe's continuity may reside not in the unchanging persistence of a particular state, but in the recursive organization through which reality continually transforms.