The Geometry of the Regime Space: How the Strategic-Inattention Boundary Bifurcates at a Cusp

Abstract

The many models of this programme are one payoff, and the boundary that sorts them into strategic-inattention and active regimes is not a single surface: past a cusp in the production response it bifurcates into a hysteretic band, and it is inside that band—on the productive substrate, at high stakes—that the programme’s dynamic traps live. An atlas (Claim A, the stage) places every model—the foundational observability-gradient equilibrium, the vertical cascade, the rotation dynamic, the single-monitor composition, the observation portfolio, its rotation and coupled extensions—as a point or region in one regime space with coordinates $(g, \delta, \alpha_0, \Delta, \Phi, \psi)$ under one of two gate substrates, sharing one local boundary $\delta^*(g, \psi)$; each placement is verified against a direct global-optimum solve. The mechanical result (Claim B) is that $\delta^*$ is a single clean boundary only for production response $g$ below a cusp curve $g_{\text{cusp}}(\delta)$; above it the boundary splits into three sheets—a lower spinodal (which is $\delta^*$ itself), a Maxwell global switch, and an upper spinodal—with hysteresis, and a single $\delta^*$ mispredicts the global regime by a gap that grows with $g$ and $\delta$. What separates the model families is not which side of a boundary they fall on but the width of the band at their operating stakes: at the weak-production corner ($\delta = 0$) the band is a razor-thin, low-stakes sliver that the programme’s static models—operating far above it—never feel, so they are effectively single-regime; on the productive substrate ($\delta > 0$) the band is wide and rises to the operating stakes, which is exactly where the dynamic (rotation) models sit. The fold is organized like a cusp—one behaviour variable (attention) folding as two controls vary, the production response $g$ and the stake $\psi$, with the production value $\delta$ sliding the pinch—but the pinch is boundary-constrained rather than a textbook interior cusp, so the catastrophe language is used as an analogy (§ 5). Its policy corollary is sharp: inside the band, recovery from the inattention trap requires crossing the lower spinodal, not the Maxwell point—the band’s width is the size of the over-correction a regime redesign must fund.