Convergence of the unfoldn-based negative binomial approximation

Prove that for all required successes r in N+ and success probabilities p in P, the expectation space negativeBinomialApprox fuel r p, defined via the kernel-based recursion unfoldn with fuel parameter and the stepNB kernel, converges to the true negative binomial distribution over the sample space Ns as the fuel parameter tends to infinity.

Background

The paper constructs an approximation to the negative binomial distribution using a kernel step (stepNB) that updates counters of required successes and observed failures, and an unfolding operator (unfoldn) with a finite fuel parameter to bound recursion. The resulting function negativeBinomialApprox : N -> N+ -> P -> & Ns yields an expectation space that approximates the negative binomial by iteratively composing probabilistic steps.

While the construction is executable and yields exact probabilities for finite fuel, the authors explicitly note that a proof of convergence to the true negative binomial distribution as fuel tends to infinity is not provided and is deferred. Establishing this convergence would validate the approximation scheme within the formal framework.