Finite-Horizon Structures IV: Local Differential Field Structures Associated with the Projective Y-Structure

This article develops the local differential field-theoretic realization of projective Y-structure within the finite-horizon framework. Starting from a smooth representative of the homogeneous scalar quantity associated with the triplet of finite-horizon structural parameters, the paper constructs the minimal local weighted structures canonically organized by the coherence 1-form attached to that representative. These include Y-compatible vector fields, weighted scalar fields, weighted tensor fields, ThetaY-twisted connections on vector bundles, and weighted divergence operators relative to a chosen reference density. The framework is purely structural and remains pre-dynamical. It does not introduce a metric, variational principle, source term, or equation of motion. Its intrinsic first-order content is carried by the coherence form, while the associated twisted connections and weighted divergence operators depend on auxiliary background choices modified by a universal logarithmic correction term. A central result is that, on the regular locus where the coherence form does not vanish, all weighted structures are locally logarithmically trivial. Weighted scalar fields reduce locally to multiples of the universal Y-weight factor, weighted fields on natural bundles reduce locally to invariant sections multiplied by the same factor, and twisted connections differ locally from background connections by a simple logarithmic correction in adapted coordinates. The article therefore identifies the minimal local operator-theoretic infrastructure induced by projective Y-structure, without yet introducing a complete dynamical theory. In this sense, it extends the finite-horizon program beyond its axiomatic, geometric, and measure-theoretic layers into a coherent local theory of weighted fields and operators. This article is self-contained at the local differential level while directly extending Finite-Horizon Structures I–III. Within the broader Ranesis program, it provides the fourth structural layer of the finite-horizon architecture: the local field-theoretic realization of projective Y-structure.