Spherical Structure and Reasoning: Geometry, Entropy, and Least-Change Trade-Offs

This record combines two small synthetic follow-ups on geometry, predictive uncertainty, and internal state-change cost. The earlier study tested synthetic point-cloud shape classification, spherical-harmonic fits, and spectral entropy: the shape classifier failed for most nonuniformly sampled spheres, and a harmonic fit alone did not establish spherical geometry. The new study comprises 72 small recurrent-network runs across two architectures, two synthetic tasks, three penalty strengths, three paired seeds, and clean or noisy training labels. A penalty on squared hidden-state updates reduced that defined update statistic, but its effect on held-out accuracy depended on task and architecture; modular addition remained near chance. Predictive output entropy was measured, not manipulated, so these experiments do not establish that entropy or data causes an optimal trade-off. Effective rank is a covariance-concentration measure, not a topology or curvature score. No real neural-network activation geometry was tested, and these results do not prove that intelligence is spherical, that anti-entropy causes intelligence, or that the proxy is physical least action. The PDFs provide the reports and references; the ZIP files provide code and data for reproduction.