Structural Potentials Under the Persistence Constraint Gradient Topology, Irreversibility, and Recovery Resistance in the LP Phase Space

Paper 000 derived the minimal admissibility architecture of determinate persistence. Paper 001 established universal real instantiation. Paper 002 derived the LP phase space. Paper 003 derived the admissibility gradient structure that constrains all persistence trajectories. The present paper derives the topology of those trajectories. Not all positions within the LP phase space are structurally equivalent. A system at IR = 0.4 and a system at IR = 0.8 are not simply located at different distances from the same boundary — they inhabit structurally different neighborhoods with different recovery capacities, different admissible trajectory spaces, and different sensitivity profiles. The phase space possesses a structural topology that is not uniform. This paper establishes five results. First, the LP admissibility structure generates structural potentials — position-dependent structural properties that describe how strongly admissible trajectories favor or resist movement toward or away from the Coherence Zone interior. Second, reserve depth generates non-equivalent structural neighborhoods: deep reserve positions admit broader recovery corridors than shallow reserve positions. This is reserve topology. Third, recovery trajectories contract as boundary proximity increases. As IR → 1, the admissible recovery neighborhood shrinks. This is recovery narrowing — the formal structural basis for the observation that crises become progressively more constrained. Fourth, the LP phase space exhibits structural hysteresis: the degradation path and the restoration path through the same IR value are structurally non-identical. The system carries structural memory of the path taken. This is not metaphorical. It is a consequence of the asymmetric trajectory structure established in Paper 003. Fifth, structural irreversibility is a formally definable condition: it occurs when the admissible restoration neighborhood collapses below the minimum reconstruction threshold — when no admissible path back to the Coherence Zone interior exists. Together these results establish that the LP phase space possesses not only geometry and dynamics but a structural topology of reversibility and irreversibility.