  We study stability of the spectral gap and observable diameter for
metricmeasure spaces satisfying the RCD(1, $\infty$) condition. We show that if
such a space has an almost maximal spectral gap, then it almost contains a
Gaussian component, and the Laplacian has eigenvalues that are close to any
integers, with dimension-free quantitative bounds. Under the additional
assumption that the space admits a needle disintegration, we show that the
spectral gap is almost maximal iff the observable diameter is almost maximal,
again with quantitative dimension-free bounds.
