K3 atoms of the cubic fourfold and the BPS structure of the Painlev\'e I determinant line

Abstract

The compatibility of the semiorthogonal decomposition of a cubic fourfold with Bridgeland stability beyond a generic point is the missing ingredient in the conjectured dynamical protection of the K3 atom $\AX$. We analyze it in the exactly solvable models of the noncommutative minimal model program. In the uncoupled Fano models ($\PP^1$; $\PP^1 \times \PP^1$ at resonance) the quantum cohomology path exits the geometric chamber at an explicit finite time, enters the selection region of the gluing, and never leaves; perturbing the resonance shows the $\varepsilon$-crossover is not a wall. The $A_2$ quiver provides a coupled counterpart. We examine the deformed cubic oscillator, Painlev\'e~I, and formulate a tau-JLO dictionary as a determinant identification and establish its perturbative layer, including computing closed-form quantum periods through $Z_4$, all-orders flatness with exact curvature the tau-divisor current, identifying it with the zeta determinant's zero divisor along a Painlev\'e~I trajectory. We close by conjecturing that the determinant's nonperturbative jumps are automorphisms of the underlying Joyce structure.