Preserve noise shared by a query and packet endpoint
The native step derivatives are qualified. Complete-history propagation also needs a covariance operator that preserves the dependence between a partial position query and the endpoint driven by the same measured increment.
Add an explicit joint-observation mode to the existing shared innovations component. For previous state x, independent standard drivers z and e, declare
observed error = H x + D z + R e
next state = F x + B z
The same z appears in both equations. The existing after-transition mode must keep its meaning and remain the default. It is the special case with observed loading H_after F and shared loading H_after B. Bind the mode into checkpoint configuration and reject schema/column mismatches.
Construct the innovation covariance and the cross-covariance with the endpoint from these shared loadings. Use the generalized Joseph update, retaining positive-semidefinite covariance through loading products, without jitter or eigenvalue clipping. Apply the same transformation to vectors and all 27 matrix columns. Future means and marginal covariance must still use earlier errors only; unknown later innovations remain random.
Compare with independent dense driver-to-observation matrices and Gaussian conditioning. Include a packet example with a partial displacement and endpoint sharing one increment, as well as changing transitions, cross-axis observation noise and positive/negative shared couplings. Demonstrate that making the shared noise independent produces a different, incorrect covariance.
Retain all 46 earlier controls. The eight new controls use at most eight generated epochs; the existing 64,080-epoch storage case stays unchanged. Use one numerical thread, with no random draws, native calls, fits, original reads or protected reads. Preserve earlier source snapshots and require a fresh source/environment-bound qualification before publishing this extension.
This qualifies covariance algebra. Composing the actual native packet derivatives over complete histories, adding persistent receiver errors and including fitted-calibration uncertainty remain separate gates before aircraft fitting. Existing accuracy limits and failed noise controls remain unchanged.