When Gaussian kernel spaces are invariant under stochastic dynamics: transition rigidity, a Gaussian-component noise boundary and a discontinuous diffusion limit

Unrefereed research candidate. Kernel methods for stochastic Koopman operators (such as kernel EDMD) often assume that the dynamics map a Gaussian reproducing kernel Hilbert space (RKHS) into itself. This candidate determines when that can happen. Rigidity. If a Markov transition with Gaussian conditional laws N(F(x),Q(x)) (arbitrary Borel fields) maps one global Gaussian RKHS boundedly into another, then Q is constant, F is affine with full-column-rank matrix M, and MRMT ⪯ P+4Q exactly. For additive noise whose characteristic function has a Gaussian lower bound (every infinitely divisible law, pure-jump included), the drift must be affine, and only the Gaussian covariance component enters the matrix boundary. Discontinuous diffusion limit. A compound-Poisson approximation of a stable linear diffusion converges in operator norm from the RKHS to bounded continuous functions at sharp order ε2, yet at t=1/10 every approximant sends the fixed observable exp(-8|x|2/5) outside the RKHS. Ornstein-Uhlenbeck answer. All-time invariance holds exactly when AP+PAT ⪯ 4BBT; the sufficient condition AP+PAT ⪯ 2BBT of Philipp et al. (2025, PAMM), whose necessity they left open, is not necessary (stable controllable shear A=[[-1,5],[0,-1]], B=P=I). The fixed affine criterion is a specialisation of classical Gaussian-Fock operator theory (Neretin); the witness and a proposed derivation were posted first, explicitly unproven, in GitHub issue the-omega-institute/trureturing #23376 (28 October 2026). Consequences: an exact bandwidth rule (isotropic width up to 4/λmax, twice the earlier range), the invariance hypothesis of an existing kEDMD error bound for multivariate OU, sampling thresholds and growth and finite-rank statements in the kernel norm. Limits: statements concern the global RKHS norm, not pointwise prediction or finite-data learning; state-dependent non-Gaussian transitions remain open. The archive contains the manuscript (PDF and Markdown source), code/verify.py (explicit checks valid under python -O), its retained output, the figure, novelty reports and SHA-256 sums. Assurance: written proofs; internal AI-assisted review only (not human peer review, independent reproduction or formal verification). Creator: Anonymous. Text CC0-1.0; code MIT.