This article develops a local theory of geometric and algebraic fields induced by a single positive homogeneous invariant defined on a smooth manifold. Starting from a canonical logarithmic one-form naturally associated with the invariant, we construct a unified framework including homogeneous scalar fields, tensor fields, compatible vector fields, derivations, connections, and divergence operators. All transformation laws are governed by the same underlying structure and do not rely on any metric, dynamical equation, or physical interpretation. The theory shows that these field structures exist on any smooth manifold, are locally gauge-trivial, and admit globally consistent formulations. Compatible connections are shown to exist on arbitrary vector bundles and to be unique up to the addition of endomorphism-valued one-forms. The resulting framework provides a minimal and universal notion of local field theory in which scaling, coherence, and integration structures are generated by a single invariant quantity. It supplies a purely geometric and categorical infrastructure for formulating scale-dependent or coherence-dependent theories independently of any specific physical model.
Alexandre Ramakers (Sat,) studied this question.