Transformations between Structural and Spectral-Temporal Representations in Generative-Relational Governance

Abstract

This paper develops a transformation framework connecting structural and spectral-temporal representations of governance within a generative-relational approach. Building on two previously distinguished representational coordinates, Type-I governance describes intervention through structural objects such as rules, dynamical processes, relational structures, and generative backgrounds, while Type-II governance describes intervention through temporal objects such as timescales, spectral components, phase relations, synchronization, resonance, polyfrequency organization, cross-frequency coupling, and spectral regimes. The present paper studies the relations between these representations rather than introducing an additional taxonomy. Structural-to-temporal transformation is modeled as a generative chain in which structural configurations produce trajectories that become observable through measurement and are subsequently represented in spectral-temporal form. Because distinct structural systems can generate equivalent temporal representations, the inverse relation is generally set-valued and gives rise to representation-relative equivalence classes and problems of structural identifiability. The framework further develops induced temporal operators, structural lifting of temporal targets, interventional identifiability, local tangent-space transformations, structural-to-temporal sensitivity maps, approximate reconstruction, and information preservation across representations. Particular attention is given to conditions under which a structural intervention induces a well-defined temporal transformation and to the multiplicity of structural realizations capable of achieving the same temporal objective. The resulting framework establishes a representational theory of generative-relational governance in which structural and spectral-temporal descriptions are complementary, partially transformable, and generally non-invertible. It provides a foundation for analyzing how governance knowledge, intervention design, and empirical inference change across representational domains.