Fortuity and fragility in supersymmetric SYK

In the N=2 supersymmetric SYK model with N fermions, every BPS state is fortuitous: at fixed charge, it exists only over a finite range of N. Allowing the charge to increase, we show that BPS classes may instead be uplifted along lattice walks inside the fortuity window, potentially to arbitrarily large N. However, there is no canonical uplift. We therefore introduce a decoder D whose spectrum quantifies exact uplift and its metric cost. We call the failure or increasing cost of uplifting harmonic representatives metric fragility. Chen's single-matrix model and the protected tower of the two-flavor SYK model realize bare and genuinely dressed mechanisms of perfect uplift, respectively. In the generic one-flavor model, exact uplift eventually fails. The Lin-Maldacena-Rozenberg-Shan-type operator D D exhibits level statistics consistent with the Gaussian unitary ensemble. Its eigenvalues quantify the metric continuity of BPS states across system size, while their correlations probe chaos within the BPS sector. Metric fragility and BPS chaos are thus encoded in complementary observables of the same operator, distinguishing chaotic BPS sectors from exactly solvable towers.