A Universal, Provably Uniform Conditioning Framework via Negative Binomial Convergence

Existing randomness conditioning methods face a fundamental dilemma: one must choose between heuristic post-processing, which is practical but lacks rigorous mathematical uniformity guarantees, and provable seeded extractors, which offer information-theoretic guarantees but require an independent perfectly uniform seed—thereby shifting, rather than eliminating, the underlying trust assumption. Neither option alone achieves provable uniformity from a raw physical source. 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 Np∼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 ZR, with spectral radius ρNB=p/p2+4(1−p)sin2(π/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 (Hin=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 χ2=275.95 (df = 255). This establishes the first seedless, provable, and platform-agnostic conditioning framework for certified entropy sources.