From States to Processes

This work continues and develops the theme introduced in "Einstein's Equation as a Correspondence Relation: Geometric and Substantial Solutions of the Problem of Interaction Between Two Localized Objects" [1]. In that preceding study, Einstein's equation was considered not only through its familiar causal mnemonic, but also as a condition of correspondence between geometric and substantial tensorial descriptions of one observable process. The geometric tensor Gμν[g] arises from metric and connection structure, whereas the stress–energy tensor Tμν arises as a macroscopic representation of a more detailed substantial state field. The present paper develops the next step of this programme. Its starting point is an asymmetry in the standard formulation of general relativity. The geometric sector is retained as a generative structure of metric, connection, curvature, geodesics, global regimes, and matching conditions. By contrast, the substantial sector is usually represented by the stress–energy tensor, which describes a body as a macroscopic whole. We propose that Tμν be understood not as the full bearer of substantial reality, but as the result of a classical projection of a richer structure of states. It is shown that an attempt to begin directly with an extended "vector" of generalized variables naturally leads to objects of different mathematical types: scalars, vectors, tensors, forms, phase regimes, correlation structures, and boundary data. Yet a global linear structure for such states should not be assumed in advance: two arbitrary states may be distinct without their distinction being expressible as a difference. A more general initial language is therefore proposed in the form of a category of substantial states. Its objects represent regimes of organization, its morphisms represent admissible processes, and the composition of morphisms represents successive transformations. Ordinary scalars, vectors, tensors, bilinear forms, and the classical stress–energy tensor then arise as local or reduced images of this structure. The purpose of this text is not to present a completed fundamental theory of substance. Rather, it records the minimal requirements for a further construction: a category of states, admissible processes, system composition, localization, boundaries, environment, local linearization, and classical and geometric projections. Particular attention is given to why the path of reasoning forms part of the result: it identifies which strong assumptions of vectorial and tensorial language must be suspended before a particular dynamics is chosen.