A Continuous Calculus of Self-Redimensioning Relational Fields: A Formal Framework for Recursive Participant, Relational, and Construct Emergence

This work proposes a formal framework for representing dynamically evolving systems as recursive relational constructs in which participant states, relational transformations, accumulated relational states, and temporal states are jointly represented. The framework introduces five central objects: a construct C, participant state P, relational function R, accumulated relational state H, and temporal state t. Participant states are represented recursively as P_n^r, where n denotes participant-domain realization and r denotes recursive state realization within that domain. This distinction permits intra-domain recursive transformation, P_n^r -> P_n^(r+1), to be separated from domain-transition emergence, P_n^r -> P_(n+1)^0. The framework investigates whether such domain-transition emergence can arise endogenously when relational transformation produces a nonzero displacement in a participant-domain coordinate already instantiated within P, rather than requiring the successor domain to be externally specified. Relational events are modeled as recursively evolving transformations whose effects may contribute to accumulated relational state and effective temporal state. Under suitable conditions, the temporal contribution of a relational event may decay toward zero, permitting participant-domain lifetimes to emerge from relational dynamics rather than being imposed as fixed parameters. The framework therefore distinguishes relational decay from participant-domain emergence and identifies the coupling conditions required for decay to influence participant transformation and successor-domain generation. The manuscript formally defines the framework's principal objects, recursive participant instantiation, relational transformation, accumulated relational state, temporal state, participant transformation, participant-domain lifetime, and construct evolution. It establishes a structural proposition for endogenous participant-domain transition conditional on a nonzero projection of participant transformation onto the domain coordinate, while explicitly distinguishing this proposition from stronger hypotheses concerning whether such a transition necessarily arises from recursive relational dynamics.A computational validation program is proposed using static relational baselines, isolated relational events, recursive decay, repeated relational events, null-model comparisons, and externally supervised adaptive-agent systems. The latter provides a potential application in which a system-level evaluator observes construct state and modifies relational dynamics across successive system realizations. The framework is presented as a theoretical model whose unresolved claims require formal proof, numerical simulation, empirical evaluation, and independent scholarly verification.