Finite-Horizon Structures V: Projective Dynamics Induced by the Y-Structure

This article develops the minimal dynamical realization of projective Y-structure within the finite-horizon framework.

Starting from one-parameter families of projective Y-morphisms, the paper defines projective Y-dynamics as compatible evolutions acting on a chosen representative of the underlying projective structure by a positive time-dependent scaling factor. A central result is that the semigroup property forces this factor to satisfy a multiplicative cocycle law, and that under a natural continuity assumption this law takes an exponential form governed by a single structural scaling exponent.

In the smooth setting, the article shows that the infinitesimal generator of a projective Y-flow satisfies the corresponding infinitesimal scaling relation and preserves the coherence 1-form attached to the projective Y-structure. This proves that the dynamical layer is fully compatible with the intrinsic first-order differential content established in the earlier geometric formulation.

The paper further establishes compatibility with the weighted local structures previously introduced, including weighted scalar fields, weighted tensor fields, and ThetaY-twisted connections. It also shows that projective Y-dynamics is compatible with the associated Y-measure class: at the intrinsic level, the measure class is preserved under suitable compatibility of the flow with the reference measure class, while at the level of chosen measure representatives the induced scaling law depends jointly on the projective scaling factor and on the pushforward scaling of the reference measure.

The framework remains purely structural and does not assume a physical interpretation, variational principle, or unique equation of motion for the triplet itself. Optional admissibility constraints, such as persistence regimes or structural thresholds, may be introduced as supplementary restrictions, but they are not part of the minimal dynamical core.

This article is self-contained at the dynamical level while directly extending Finite-Horizon Structures I–IV. Within the broader Ranesis program, it provides the fifth structural layer of the finite-horizon architecture: the theory of compatible projective evolution.