What the Data Can See: Sheet-Specific Information Signatures of the Strategic-Inattention Fold

Abstract

The strategic-inattention trap studied by a companion regime-space analysis presents, as a stakes parameter is varied, a fold with three distinct sheets: a boundary collision at which the trap configuration is born — the recovery boundary, below which a trapped unit returns to engagement — a Maxwell sheet at which the engaged and trap configurations carry equal payoff, and an interior saddle-node at which the engaged configuration is destroyed. This paper asks what that fold looks like from data, and shows that the three sheets carry three distinct information signatures whose visibility is governed by the observation window. Short-window, branch-conditional data — the regime into which finite audit samples fall, because the trap is an absorbing state with exponentially long escape times — carry identifying power that diverges at the interior saddle-node. Long-run ensemble statistics, including a rising variance, peak instead at the Maxwell sheet. The boundary collision is the recovery boundary — the stakes level below which the trap ceases to be a rest point and a trapped unit can return — and it is informationally silent: on the trap branch the observed action does not respond to the stakes, so branch-conditional Fisher information about the parameters that set the recovery boundary is exactly zero. The fall itself is not the hidden event. A unit tracking the engaged optimum departs adiabatically at the interior saddle-node, where branch-conditional identification diverges, and the classical critical-slowing-down variance of its fluctuations diverges there too; shock-driven falls, which can occur anywhere in the band, register in the ensemble weight. What no monitoring statistic reveals is recovery: whether a trapped unit could be brought back is decided by the very parameters its silent data cannot identify. The early-warning reading must also be qualified, because a rising variance is ambiguous between two observables on two sheets — the between-basin ensemble variance, which peaks at the Maxwell sheet, and the within-well fluctuation variance, which grows toward the saddle-node — so that reporting a rising variance without saying which mislocates the transition. The upshot is a reversal of the naive worry: the fall is visible, and it is recovery that is not. The signature locations are shown to be robust across the detection sensitivity and the production value and across two distinct short-window observation models, while their exponents and magnitudes are calibration-dependent; the finite-noise correction to the ensemble signature is not merely bounded but predicted by a two-well prefactor law. A remark notes that the penalty scale — the parameter policy sets most directly — is the flattest direction of the model manifold, and hence the one monitoring data least identify.