This research release studies a precise question about polynomial exponential periods: when does one chosen integrand stop generating the full differential system as its parameters change? The candidate turns repeated Gauss-Manin differentiation into a finite exact matrix. The minors of that matrix locate parameter values where the chosen seed loses rank. For a depressed quartic phase, the paper gives a complete rank stratification and distinguishes a defect that persists along the even family from another that is generally only a pointwise failure of one derivative frame. For an odd quintic, the package supplies a finite exact certificate containing the full matrix, four reduction witnesses, the determinant, all sixteen size-three minors, and explicit rank-one, rank-two, rank-three, and rank-four regions. The release also uses the known Chebyshev-Bessel family as a regression benchmark; it does not claim discovery of that integral or differential equation. The public evidence package includes the manuscript, source code, machine-readable claims, hashes, replay receipts, citation checks, deliberate formula mutations, and invalid-input controls. All packaged checks pass from fresh extracted copies under multiple Python versions. This remains an anonymous, unrefereed candidate. It has not been independently reproduced, formally verified, reviewed by a specialist, or conventionally peer reviewed, and novelty or priority is not established. This is an Evidence Press audio briefing, released 8 August 2026. The full paper, exact evidence package, and open follow-up problems are linked on this page. The audio uses an AI-generated voice and is not additional mathematical evidence.