Gaussian-kernel plug-and-play iterations can be unstable: a sharp dimension threshold for irreducible unit-diagonal kernels, the role of the mixing gap, and why unit diagonals matter

Unrefereed research candidate. Plug-and-play (PnP) methods for linear inverse problems alternate a data step with a denoiser. Sinha, Kumar, Athalye and Chaudhury (IEEE Trans. Signal Process. 73, 2025, Section II-A) observed that PnP-ISTA with row-normalised kernel denoisers, such as Gaussian non-local means, seems to converge in practice but that they had found neither a proof nor a counterexample. This package answers the question for PnP-ISTA and PnP-ADMM.

With irreducible unit-diagonal kernels and a data Hessian that preserves constants, both iterations are stable in dimensions one to three for every admissible step. In dimension four they need not be: explicit Gaussian kernels with a fixed positive forward operator make both diverge at any prescribed step up to one half, and an exact two-parameter classification locates the boundary (t* ≈ 0.5313). A twelve-dimensional positive convolution example fails exactly for 0 < t < 1.9157×10⁻⁴. Without unit diagonals, failure already occurs in dimension three. The cause is a small mixing gap: steps below g/√κ_π are always safe and linear scaling in g cannot be improved. Symmetric denoisers are stable exactly when no vector fixed by the denoiser is annihilated by the Hessian, extending Athalye, Chaudhury and Kumar (2023).

Practical size: the divergence is extremely slow (at most about 1.7×10⁻⁵ per iteration, so tenfold growth needs 10⁵–10⁶ iterations) and the affinities involved are tiny; the results are structural and do not demonstrate divergence in an imaging pipeline. Not the first disproof of the broader Banerjee–Chaudhury conjecture (S. Patel's online counterexample came first).

The archive contains the 28-page manuscript (PDF and Markdown source), exact-arithmetic verifiers, coefficient files and a standard-library checker for the Theorem 3 and 4 certificates (printed in Appendix C), semantic negative controls, a replay driver, CI and SHA-256 sums. Assurance boundary: written proofs; exact rational certificates; producer-side replay from a clean copy; internal AI-assisted review (multi-perspective review, correctness audits by a different model, a targeted proof audit and a verification re-review). This is not peer review, independent reproduction or formal verification. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.