Tunderives II: Information Hierarchy, Grassmannian Trajectories, and Cross-Domain Geometric Coherence

Tunderives II presents a geometric framework for exploring how organization becomes observable in dynamical systems through successive levels of representation. The central idea is that observability depends on geometric resolution. Starting from a measured signal, information is transformed through a sequence of deterministic representations, culminating in geometric trajectories and observable structures. Within this framework, all observables are viewed as different projections of the same underlying trajectory, providing a unified perspective on quantities that are often studied independently. The manuscript introduces an Information Hierarchy linking signal representations, Grassmannian trajectories, spectral organization, and observable correlations. A common entropy operator is used to define complementary quantities acting at different descriptive levels, creating a bridge between local geometric structure and higher-level observables. Several geometric observables are revisited, refined, and verified through a process of correction, testing, and validation. Exploratory empirical analyses are presented on biological and engineering datasets, including algae raceway oxygen measurements and NASA lithium-ion battery degradation data. These studies illustrate how geometric representations can reveal reproducible structures that are not immediately visible in raw signals. Rather than proposing a final theory, Tunderives II introduces an observational framework designed to support further investigation across domains. The work is motivated by the possibility that coherence may emerge progressively across multiple geometric resolutions, and that different representations may reveal complementary aspects of the same underlying dynamics. The framework is intended as an instrument for exploration: a reproducible geometric lens through which organization, structure, and coherence may be studied across complex systems. The horizon remains open, inviting future work to investigate how observability evolves as geometric resolution increases and new representations become accessible.