Formal Relational Dynamics (FRD): A Framework for Dynamical Systems on Evolving Metric Structures

We introduce Formal Relational Dynamics (FRD), a mathematical framework in which dynamical systems are defined not on a fixed metric space, but on a time-evolving relational structure. The central object is a relational space (Rₜ, muₜ, cₜ), where muₜ is a probability distribution over configurations and cₜ is a state-dependent ground cost inducing a Wasserstein-2 metric. A relational operator Tₜ simultaneously updates the distribution and the metric, so that geometry emerges from dynamics rather than being imposed externally. Key results: (1) Irreversibility is proven structurally from the non-invertibility of the metric update, without any entropy postulate. (2) The 3D logarithmic spiral is identified as the first concrete FRD realization, with a proven critical scale s* at which the spiral transitions to a helix — the FRD fixed-point attractor. (3) A geometric invariant G = omega/ (2 z₀) is conjectured and numerically verified to less than 0. 1% across a parameter range. (4) Metric asymmetry of approximately 36% is demonstrated numerically, confirming irreversibility in the geometric realization. Five open problems are stated precisely. This paper is the mathematical foundation of a series: the Thermodynamic Manifold Framework (TMF) and Climate Manifold Dynamics (CMD) are physical realizations developed in companion papers. This is a working paper / preprint; results are marked as PROVEN, CONJECTURE, or OPEN throughout.