This article develops a minimal differential-geometric framework associated with finite-horizon structural invariants, extending an earlier axiomatic and categorical formulation. Starting from structured objects described by three strictly positive parameters—interpreted abstractly as magnitude, coherence, and evaluation horizon—the work considers the situation in which these parameters vary smoothly across a manifold. Without introducing any physical, dynamical, or metric assumptions, it investigates which geometric structures are unavoidably induced by the homogeneity of the underlying invariant alone. The central result is the construction of a minimal geometry entirely generated by the invariant’s infinitesimal variation. This geometry gives rise to a canonical coherence one-form, a rank-one degenerate pseudometric, a transverse notion of distance, and a family of compatible torsion-free affine connections. The resulting structure is intrinsically flat and naturally foliated by equivalence classes of constant structural balance. A key feature of the framework is that all non-trivial geometric content is confined to a single transverse direction, while variations tangent to the foliation are structurally neutral. The geometry therefore encodes structural differentiation rather than geometric distortion. The article further establishes an exact correspondence between structure-preserving morphisms in the underlying abstract category and isometries of the induced transverse geometry. In this way, categorical equivalence is translated faithfully into differential-geometric equivalence. The framework is deliberately pre-dynamical and model-independent. It does not introduce equations of motion, physical time, curvature dynamics, or empirical interpretation. Instead, it provides a differential realisation of finite-horizon structural identity, serving as a stable geometric interface for later dynamical, higher-order, or application-oriented developments within the Ranesis program.
Alexandre Ramakers (Tue,) studied this question.