This article constitutes the sixth volume of the Finite-Horizon Structures series and extends the Ranesis framework based on the positive homogeneous invariant at the core of the theory. Volumes I–V established the axiomatic, categorical, differential, measure-theoretic and dynamical foundations induced by this invariant. The present work builds directly on those results to develop a purely geometric theory of superlevel regimes and structural transitions. Introducing the notions of local fertility, binary fertility invariant, and fertility spectrum, the article proves the existence of an intrinsic transition level separating fertile structural regimes from sterile ones. This threshold is not imposed externally: it emerges entirely from the invariant, its canonical differential structure, the induced Radon measure, and a compatible aggregation operation. Within the Ranesis program, this result provides a fully intrinsic and non-probabilistic reformulation of percolation phenomena. Structural transition appears as a geometric property of invariant-induced filtrations and remains stable under structural equivalence. This volume therefore integrates percolation-like behaviour into the global architecture of Ranesis, without invoking graph theory, lattice models, or probabilistic assumptions.
Alexandre Ramakers (Sun,) studied this question.