When Wrong Answers Remain Unsettled: Failure Dynamics in Recurrent Reasoners

Weight-shared recurrent models allocate additional computation by iterating the same blocks, but a wrong terminal answer need not be a settled one. We measure answer persistence under continued recurrence in a trained Hierarchical Reasoning Model (HRM; 27M parameters, Sudoku-Extreme), comparing perturbed continuation with an unperturbed control. After one additional inference budget, mild wrong answers persist at a rate of $0.42$ ($n=170$), versus $0.95$ for solved endpoints ($n=512$); the rate pooled over $512$ wrong endpoints is $0.18$. Perturbations spanning $0.05$–$20\times$ the terminal-update scale have little effect on these rates. The measurement therefore primarily reflects finite-horizon answer persistence, rather than sensitivity to perturbation or asymptotic attraction. Solved HRM endpoints have a mean leading-Ritz radius of $0.44$, while scheduled linear models predict fast-loop updates better than slow-loop updates (held-out $R^2=0.72$ versus $0.37$). We also evaluate established perturb-and-reselect procedures with paired gains, regressions and oracle headroom. The frozen noise recipe gains $4.8$ accuracy points ($197$ gained, $2$ lost); compute-matched extended depth gains $5.6$, and Sudoku symmetry augmentation gains $34.6$. Across six evaluated settings, solved answers remain more persistent than wrong answers, but the fixed persistence threshold and HRM spectral and timescale patterns do not generalise universally. TRM and GRAM-guided TRM share a module across their two levels. The timescale ordering reverses on unsolved TRM states and in the GRAM-guided setting; in TRM both injection sites give similar gains. These results separate terminal-answer persistence, local geometry and reselection efficacy, and delimit which failure diagnostics transfer across recurrent models.