When canonical covariance localisation fails: Exact finite-ensemble counterexamples, a sharp leading-order criterion, and the Snyder–Hakim test case

Unrefereed research candidate. In one sentence: whether Snyder and Hakim's canonical coordinates are the best variables in which to localise an ensemble covariance depends on how analysis error is scored; under physical squared error the conjecture fails at every ensemble size, while under prior-normalised (Mahalanobis) error it holds at leading order. Ensemble Kalman filters estimate forecast uncertainty from a small ensemble and localise the noisy sample covariance. Snyder and Hakim (JAMES 2022) conjectured that canonical coordinates (whiten the prior and observation errors, then rotate to the singular vectors of the whitened observation operator) are the best basis in which to localise. Setting: one analysis step, an independent Gaussian ensemble of size N, an oracle (known) prior covariance used to build every basis, and deterministic Schur-product tapers. Physical squared error: for one scalar observation whose direction is not an eigenvector of the prior covariance, an exact first-variation formula shows that at every N ≥ 2 a sufficiently small rotation followed by complete localisation beats every canonical taper with unit observed diagonal; with the same deterministic variance rescaling it also beats every canonical taper with any observed-variance factor. Two two-dimensional examples are certified in exact rational arithmetic; in one the canonical basis is a strict local maximum of the risk. With several observations, coupling between an observed and an unobserved canonical direction makes canonical complete localisation non-stationary at every N (compared with rotated complete localisations). Large ensembles: a uniform risk expansion gives the leading basis-dependent objective F(U) = ∑i(uiTBui)(uiTAui), and a sharp criterion says exactly when canonical coordinates minimise it. Under prior-normalised loss they always do, and for a scalar observation canonical complete localisation is a strict local minimum at finite N provided a scalar ζ is positive (numerically supported, unproved). In a reconstruction (not a replication) of Snyder and Hakim's 200-variable test case, canonical coordinates leave about 42–54% of the avoidable excess error; symmetric (ZCA) whitening does almost as well as the best basis found, roughly equivalent to 1.7–2 times more ensemble members; bases estimated from small pilot samples keep much less of the gain. The classical trace inequality used is credited to Bourin and Fujii; the canonical-taper lemma is the zero-correlation case of Flowerdew (2015). The two-observation sign conjecture and the prior-normalised finite-N inequality are numerical only. The archive contains the manuscript (PDF and LaTeX source), the bibliography and its Crossref-based generator, figures, four verification scripts (exact rational certificates; deterministic quadrature; theory checks with a 4 × 106-ensemble Monte Carlo and negative controls; the testbed reconstruction) with their recorded outputs, a claim-by-claim verification table, a rating-free internal review summary, development notes, CI and SHA-256 sums. Assurance: written proofs, not machine-verified; exact-arithmetic and floating-point evidence kept separate; internal AI-assisted review only (not peer review, independent reproduction or formal verification); bounded, dated literature search. Creator: Anonymous. Text CC0-1.0; code MIT.