Finite-Horizon Structures II: Differential Geometry Induced by the Projective Y-Structure

This article develops the minimal differential-geometric realization of projective Y-structure in the smooth setting. Starting from smooth positive representatives of the structural triplet associated with finite-horizon organization, the paper constructs a canonical first-order differential framework centered on the coherence 1-form attached to the homogeneous scalar field determined by the triplet. From this form, it derives a rank-one degenerate symmetric tensor, a natural pseudometric distance on path-connected components, and a regular foliation on the locus where the differential content is non-vanishing. The framework is deliberately minimal and pre-dynamical. It does not introduce physical interpretation, equations of motion, or additional metric input. Instead, it identifies the weakest differential-geometric structures forced by smoothly varying projective Y-organization alone. A central result is that the intrinsic first-order differential content is canonically flat in the de Rham sense: the coherence form is exact, and no non-trivial curvature tensor is determined by the minimal data alone. Auxiliary torsion-free affine connections compatible with the coherence form may nevertheless be introduced on the regular locus, providing a notion of parallel transport and autoparallel curves without becoming part of the intrinsic foundation. The article also establishes the precise compatibility between the smooth geometric framework and the axiomatic category introduced in Finite-Horizon Structures I. Geometric Y-morphisms preserve the coherence form and the degenerate Y-metric, while geometric Y-isomorphisms act isometrically on the induced transverse pseudometric geometry. This work is self-contained at the geometric level, while directly extending the categorical and projective framework of Finite-Horizon Structures I. Within the broader Ranesis program, it provides the minimal differential layer upon which later geometric, dynamical, and domain-specific developments may be built.