Independent Information from Poisson Observations: Spectral Classification and Compatibility

This preprint studies what information a single Poisson count of unknown intensity can provide alongside a prescribed observation. One randomized rule must produce the target and a second output that is independent of it at each intensity. For the full final state of a finite irreducible Markov chain, informative independent returns are characterized by positive mass in every cyclic class and positive real parts of the active peripheral eigenvalues. A spectral obstruction also applies to eventually periodic targets, with a converse when every occurring output has positive probability in each tail row. A periodic example shows that coarsening can restore an informative return. For a primitive two-state target, two explicit bits can be returned individually, but no admissible return can supply both. A finite rational certificate establishes this incompatibility over an explicit range of transition and thinning parameters. The paper also compares split Poisson residuals for deterministic observations and bounds simulation error at finite valuation depth. The supplement contains exact verification code and its results.