Observable systems with divergent isolated Kalman covariances: a counterexample to the isolated Kalman filtering convergence conjecture

Unrefereed research candidate. The isolated Kalman filter (IsoKF) of Jung, Luft and Weiss (Autonomous Robots, 2025) lets the nodes of a robot or sensor network share an estimation task cheaply: only the nodes involved in a measurement update, and their correlations with everyone else are rescaled by a correction factor. Its authors conjectured that for any completely observable and controllable system the covariance it reports converges to a finite, unique, stable steady state.

Result: the conjecture is false. A three-node system with pair-sum observations x1+x2, x2+x3, x3+x1 and dynamics A = 3I is observable from one epoch and controllable, every local block the filter uses stays positive definite, yet the reported marginal variances satisfy m_k ≥ (5/4)(9/5)^k − 1/4, while the centralised Kalman filter keeps every variance at most 3/4. A Perron-root comparison extends divergence to every connected non-bipartite pair-sum graph when a^2 is large enough; extra attenuation or deletion of cross-covariances does not help. Schur-stable dynamics always give bounded marginals (proved). Computer-assisted proofs (outward-rounded interval arithmetic) show that for a^2 in (1, 1.38] large priors diverge at rate a^(2k), entry certified from 10^12 I at eight values down to a^2 = 1 + 10^-8, and that at a^2 = 21/20 the prior I converges while 10^12 I diverges: there is no globally attracting steady covariance. Also: exact classification of participant overlaps (|U ∩ V| ≤ 1) that keep every local block positive definite, and a bounded regime for anchored relative measurements.

Scope: in high-precision numerical tests the actual estimation error stays bounded; the failure is gross pessimism of the reported covariance, not divergence of the estimate. The counterexample uses pair-sum measurements, not plain differences. Nothing is proved for a^2 in (1.38, 2+√5); uniqueness for stable dynamics remains open. The ikf_lib software was inspected but not run; no software failure is claimed.

Archive: LaTeX manuscript and PDF, bibliography with registry records, exact rational verifiers, interval certificates C1–C3, numerical scripts and their outputs (normal and python -O), figures, CI and SHA-256 manifest. Assurance: internal AI-assisted review only (not human peer review, independent reproduction or formal verification); certificates are computer-assisted, not formal proofs. Creator: Anonymous. Text CC0-1.0; code MIT.