ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A singular prior factor with a positive innovation remains admissible

Example

Take one state and one observation, one update step, $m_{0} = 2$ , $F = H = 1$ , $y = 7$ , and factors $L_{0} = L_{Q} = 0$ , $L_{R} = 1$ . The initial, process, and measurement covariances are $P_{0} = Q = 0$ and $R = 1$ .

Facts & Assumptions

Given: The scalar data in the Example and the exact factor update of Finite-horizon square-root Kalman filter by rectangular factors and QR.

[F1]

Prediction concatenates $F L_{0}$ and $L_{Q}$ , then pads the factor array by zero columns as needed. Finite-horizon square-root Kalman filter by rectangular factors and QR

[F2]

The measurement prearray has blocks $[\begin{array}{cc} L_{R}^{p a d} & H L^{-} \\ 0 & L^{-} \end{array}]$ . Finite-horizon square-root Kalman filter by rectangular factors and QR

[F3]

For positive innovation, the block QR factor gives the canonical posterior covariance and mean, with singular posterior factors allowed. The block QR measurement update gives the canonical Kalman update

Proof

Given: The scalar inputs in the Example.

1.1F1givenalgebra

The prediction array is $[F L_{0}, L_{Q}] = [0, 0]$ . Its QR upper factor is zero, so $L^{-} = 0$ and $P^{-} = 0$ . The predicted mean is $m^{-} = 1 \cdot 2 = 2$ . The innovation is $v = 7 - 1 \cdot 2 = 5$ , and its covariance is $S = H P^{-} H^{T} + R = 1$ . Thus the canonical innovation is positive definite even though $P_{0}$ and $Q$ are singular.

2.1F2step 1.1algebra

The measurement prearray is $Z = [\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}] .$ Its exact QR gives $L_{11} = 1$ , $L_{21} = 0$ , $L_{22} = 0$ . The solve gives $z = 5$ , so $m^{+} = m^{-} + L_{21} z = 2$ , while the represented posterior covariance is $L_{22} L_{22}^{T} = 0$ .

3.1F3step 2.1algebra∎

The canonical covariance update is $P^{+} = 0 - 0 \cdot 1^{- 1} \cdot 0 = 0$ and the canonical mean update is $2 + 0 \cdot 1^{- 1} \cdot 5 = 2$ . They agree with the factor output in step 2.1. The zero posterior factor is admissible by [F3]; no Cholesky factor of a positive-definite state covariance is required.

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