This paper formulates and solves the inverse problem of recovering the pre-resampling spectrum of a particle filter's ensemble covariance from the importance weights and the observed spectrum alone. Resampling controls weight degeneracy, but in high dimensions it also distorts that spectrum: the distortion is a free multiplicative convolution with a law computable from the weights, so the repair is a deconvolution. We first extend the forward spectral law to random importance weights by conditioning on the weights together with the realized offspring counts rather than on the weights alone, which makes the distortion law deterministic at the observed data and gives the zero-atom mass of the limiting law a direct reading as an effective-dimension proxy for the second-order information the filter has lost. Second, a regularized free deconvolution inverts the distortion with an explicit error budget at a fixed resolution scale, and a computable forward residual certifies when the model behind it is violated. Injectivity makes the inverse identifiable but not stable, so the budget rests on a resolution-restricted condition number whose positivity we prove, and is vacuous below that resolution rather than concealed in a hypothesis. Third, the repaired spectrum enters only as the preconditioner of a Metropolis-adjusted rejuvenation move, so spectral error changes the mixing constants and not the invariant target, and the full cycle obeys a geometrically summable error recursion whose abstention term is governed by when abstentions fall rather than how many occur. Pre-registered simulations confirm the forward law, an order-of-magnitude reduction of spectral error, and the abstention law. The repair recovers eigenvalue laws, not eigenvectors.

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