A Universal, Provably Uniform Conditioning Framework via Negative Binomial Convergence

Source abstract

Existing randomness conditioning methods face a fundamental dilemma: heuristic post-processing lacks rigorous mathematical uniformity guarantees, while provable seeded extractors require an independent perfectly uniform seed, merely shifting the trust assumption. We present a universal, mathematically certified conditioning framework that resolves this dilemma. For any NIST SP~800-90B ESV-certified entropy source, regardless of bias or implementation, our framework generates a provably uniform output stream without requiring any external seed or heuristic whitening. The core contribution is the Geometric Convergence Theorem (GCT), proving that the modular reduction of a negative binomial counting variable $N_p \sim \mathrm{NB}(m, p)$—where m denotes the required number of successes generated from fixed ESV entropy blocks via Bernoulli trials with success probability p—converges exponentially to uniformity over $\mathbb{Z}_R$, with spectral radius $\rho_{\mathrm{NB}} = p / \sqrt{p^2 + 4(1-p)\sin^2(\pi/R)} < 1$. A Practical Entropy Budgeting mechanism ensures information-theoretic entropy conservation via a fixed input-output ratio. In a large-scale validation generating 100 MB of output from a biased ESV source ($H_{\mathrm{in}}$ =3.32 bits/byte), the framework achieved Shannon entropy 7.999998 bits/byte and Min-Entropy 7.9936 bits/byte, approaching the theoretical lower bound of 7.9949 bits/byte to within 0.0013 bits/byte, with$\chi^2 = 275.95$ (df = 255). This establishes the first seedless, provable, and platform-agnostic conditioning framework for certified entropy sources.