A Universal, Provably Uniform Conditioning Framework via Negative Binomial Convergence

1. Introduction

The generation of certified randomness is a foundational requirement for modern cryptographic infrastructure. Quantum key distribution (QKD), post-quantum cryptographic (PQC) primitives, blockchain protocols, and secure communication systems all depend on high-quality entropy for key generation, nonces, and ephemeral parameters [1,2]. The security guarantees of these systems are only as strong as the randomness that seeds them.

Recent work on software-based entropy sources has demonstrated that high-quality randomness can be harvested from system-level phenomena such as microarchitectural timing jitter and combinatorial sorting processes. The Quantum Permutation Pad Random Number Generator (QPP-RNG) [3] showed that permutation sorting operations on commodity hardware produce raw outputs that pass NIST SP 800-90B statistical tests across diverse platforms. The Random Permutation Sorting System (RPSS) [4] provided a theoretical framework for this phenomenon, establishing that the permutation count follows a negative binomial distribution and that its modular reduction yields empirically uniform outputs. These observations raised a fundamental question: can we extract a provable mathematical principle from this physical behavior, and generalize it to any ESV-certified entropy source?

Existing approaches to random number generation fall into several categories, each with fundamental limitations:

Pseudorandom Number Generators (PRNGs) expand a small seed deterministically but rely entirely on external entropy to generate that seed. They are not self-sufficient entropy sources.

True Random Number Generators (TRNGs) and Quantum Random Number Generators (QRNGs) exploit physical stochasticity—thermal noise, oscillator jitter, single-photon detection, or vacuum fluctuations—to produce entropy [1]. However, raw physical outputs are almost invariably biased, correlated, or non-uniform, necessitating post-processing or conditioning to achieve statistical uniformity.

This conditioning step presents a critical vulnerability:

• Heuristic approaches (e.g., SHA-256 hashing, AES-based whitening, von Neumann extraction) are empirically effective but lack rigorous mathematical guarantees. They rely on computational assumptions (for cryptographic hashing) or restrictive source models (for von Neumann, which requires IID inputs) [5,6].
• Seeded extractors, such as those based on the Leftover Hash Lemma (LHL) [7,8], provide information-theoretic guarantees but require a short, independent, and perfectly uniform seed. This seed itself must come from an already-certified entropy source, creating a circular dependency that is often resolved by heuristic assumptions or additional hardware.
• Deterministic extractors—which would eliminate the seed requirement—are known to be impossible for arbitrary high min-entropy sources [9]. No single deterministic function can map every possible high-entropy distribution to uniform output.

This landscape leaves a critical gap: Is there a way to achieve mathematically provable uniformity from a physical entropy source without requiring a separate uniform seed, and without relying on heuristic assumptions? The empirical success of QPP-RNG and RPSS suggests that such a framework may exist, but it requires a rigorous mathematical foundation that extends beyond the specific hardware implementations.

1.1. The NIST ESV Framework as a Bridge

NIST Special Publication 800-90B [10] establishes a rigorous framework for Entropy Source Validation (ESV). An ESV-certified source provides a conservative, worst-case estimate of its min-entropy $H_{\mathrm{in}}$ (bits/byte), derived from statistical tests that quantify predictability under the most adversarial interpretation of the data. Similar frameworks, such as the German BSI AIS 20/31 standard [11], provide complementary methodologies for RNG evaluation, underscoring the global importance of certified entropy sources.

Critically, ESV certification provides a structural guarantee: it confirms that the entropy source is non-degenerate—that is, it possesses genuine, irreducible physical unpredictability. This guarantee, while statistical in nature, is sufficient to form the foundation for a mathematically rigorous conditioning module.

1.2. Our Contribution: A Seedless, Provable Conditioning Framework

In this work, we present a universal conditioning framework that bridges ESV certification with information-theoretic uniformity guarantees. Building on the insights from RPSS—where the permutation count’s negative binomial distribution and its modular convergence to uniformity were observed empirically—we elevate this phenomenon to a general mathematical theorem. Our approach is built on three pillars:

1.

The Geometric Convergence Theorem (GCT): For any ESV-certified non-degenerate source, the modular residue $Y=N_{p}modR$ of a negative binomial counting variable $N_p\sim \mathrm{NB}(m,p)$ converges exponentially to the uniform distribution over ${\mathbb{Z}}_R$. The convergence rate is governed by a closed-form spectral radius ${\rho }_{\mathrm{NB}}<1$ that depends only on p and R.

2.

Practical Entropy Budgeting: We provide a simple, rigorous entropy accounting mechanism: each symbol consumes $L=\lceil {log}_{2}R/H_{\mathrm{in}}\rceil +C$ input bytes, where $R=16$ is the internal symbol size and $C\ge 1$ is a configurable safety margin. This guarantees $L·H_{\mathrm{in}}>{log}_{2}R$, satisfying information-theoretic entropy conservation.

3.

Efficient Mathematical Sampling: The negative binomial counting process is realized via deterministic Bernoulli trials simulated from ESV-seeded entropy blocks. The implementation is platform-agnostic, requires no hardware calibration, and achieves high throughput through pure integer operations. The sampler can be instantiated with any deterministic PRNG (e.g., LCG, ChaCha20) without affecting the GCT proof.

1.3. Comparison with Existing Approaches

Table 1 positions our Negative Binomial Conditioner (NBC) framework against heuristic whitening, seeded extractors, and deterministic extractors. Unlike seeded extractors, our approach requires no external seed; unlike heuristic methods, it provides mathematically provable uniformity; and unlike deterministic extractors, it is applicable to all ESV-certified sources with any positive min-entropy. This combination makes it the first seedless, provable, and platform-agnostic conditioning solution for NIST-compliant entropy sources.

1.4. Paper Organization

Section 2 presents the mathematical foundations. Section 3 introduces the entropy budgeting mechanism and conditioning algorithm. Section 4 provides empirical validation. Section 5 discusses implications. Section 6 concludes.

2. Theory of Provable Uniformity

We establish a rigorous mathematical framework for transforming non-degenerate entropy into certified uniform randomness. The core object is the Negative Binomial distribution, which arises naturally when counting trials until a fixed number of successes in a sequence of independent Bernoulli experiments.

2.1. The Negative Binomial Distribution

Let ${\left\{B_i\right\}}_{i=1}^{\infty }$ be a sequence of independent Bernoulli trials with success probability $p\in (0,1)$. Define the random variable $N_p$ as the total number of trials required to observe exactly m successes:

$$N_p=inf\left\{n\ge m:{\displaystyle \sum _{i=1}^n}{B}_i=m\right\}.$$

Then $N_p$ follows the negative binomial distribution under the total-trials parameterization, with probability mass function:

$$Pr(N_p=n)=\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{m-1}}\right)p^m{(1-p)}^{n-m},n=m,m+1,\dots$$

and probability generating function:

$$G_{N_p}\left(z\right)={\left({\displaystyle \frac{pz}{1-(1-p)z}}\right)}^m.$$

The expectation is $\mathbb{E}\left[N_p\right]=m/p$ and the variance is $\mathrm{Var}\left(N_p\right)=m(1-p)/p^2$.

2.2. The Geometric Convergence Theorem

We now state and prove our main theorem: the modular reduction of a negative binomial random variable converges exponentially to uniformity. The proof employs discrete Fourier inversion over the cyclic group ${\mathbb{Z}}_R$, a standard technique for analyzing convergence to uniform distributions on finite groups [14]. This theorem generalizes the empirical observations of modular convergence in the RPSS framework [4] to any process that generates a negative binomial counting variable.

Theorem 1.

Geometric Convergence to Uniformity. Let $N_p\sim \mathrm{NB}(m,p)$ with $0<p<1$. For any integer modulus $R\ge 2$, define the spectral radius:

$${\rho }_{\mathrm{NB}}\underset{1\le r\le R-1}{max}\left|{\displaystyle \frac{p{e}^{i2\pi r/R}}{1-(1-p)e^{i2\pi r/R}}}\right|.$$

Then ${\rho }_{\mathrm{NB}}<1$, and the distribution of $Y=N_{p}modR$ satisfies:

$$\underset{0\le k<R}{max}\left|Pr(Y=k)-{\displaystyle \frac{1}{R}}\right|\le {\displaystyle \frac{R-1}{R}}{\rho }_{\mathrm{NB}}^m.$$

Proof. 

The characteristic function of $N_p$ is:

$${\varphi }_{N_p}\left(\omega \right)={\left({\displaystyle \frac{p{e}^{i\omega }}{1-(1-p)e^{i\omega }}}\right)}^m.$$

For the modular variable $Y=N_{p}modR$, the probability mass function is given by the discrete Fourier inversion formula:

$$Pr(Y=k)={\displaystyle \frac{1}{R}}{\displaystyle \sum _{r=0}^{R-1}}{e}^{-i2\pi rk/R}{\varphi }_{N_p}\left({\displaystyle \frac{2\pi r}{R}}\right).$$

The $r=0$ term equals $1/R$ since ${\varphi }_{N_p}\left(0\right)=1$. Thus:

$$Pr(Y=k)={\displaystyle \frac{1}{R}}+{\displaystyle \frac{1}{R}}{\displaystyle \sum _{r=1}^{R-1}}{e}^{-i2\pi rk/R}{\varphi }_{N_p}\left({\displaystyle \frac{2\pi r}{R}}\right).$$

Taking absolute values and applying the triangle inequality:

$$\left|Pr(Y=k)-{\displaystyle \frac{1}{R}}\right|\le {\displaystyle \frac{1}{R}}{\displaystyle \sum _{r=1}^{R-1}}\left|{\varphi }_{N_p}\left({\displaystyle \frac{2\pi r}{R}}\right)\right|.$$

For each $r=1,\dots ,R-1$:

$$\left|{\varphi }_{N_p}\left({\displaystyle \frac{2\pi r}{R}}\right)\right|={\left|{\displaystyle \frac{p{e}^{i2\pi r/R}}{1-(1-p)e^{i2\pi r/R}}}\right|}^m\le {\rho }_{\mathrm{NB}}^m.$$

Therefore:

$$\left|Pr(Y=k)-{\displaystyle \frac{1}{R}}\right|\le {\displaystyle \frac{R-1}{R}}{\rho }_{\mathrm{NB}}^m.$$

Taking the maximum over k completes the proof.    □

2.3. Closed-Form Spectral Radius

The spectral radius can be expressed in closed form.

Lemma 1.

For $0<p<1$ and $R\ge 2$,

$${\rho }_{\mathrm{NB}}={\displaystyle \frac{p}{\sqrt{p^2+4(1-p){sin}^2(\pi /R)}}}.$$

Proof. 

For $r=1,\dots ,R-1$,

$${\left|{\displaystyle \frac{p{e}^{i\theta }}{1-(1-p)e^{i\theta }}}\right|}^2={\displaystyle \frac{p^2}{1-2(1-p)cos\theta +{(1-p)}^2}}.$$

Using $1-cos\theta =2{sin}^2(\theta /2)$, the denominator becomes:

$$1-2(1-p)cos\theta +{(1-p)}^2=p^2+4(1-p){sin}^2(\theta /2).$$

The maximum occurs at $\theta =2\pi /R$, yielding the stated expression.    □

With our recommended parameters $p=0.04$, $R=16$, and $m=4$:

$${\rho }_{\mathrm{NB}}={\displaystyle \frac{0.04}{\sqrt{0.0016+4\left(0.96\right){sin}^2(\pi /16)}}}\approx 0.1041.$$

The deviation bound is:

$${\displaystyle \frac{15}{16}}\times {\left(0.1041\right)}^4\approx 1.10\times {10}^{-4}.$$

2.4. Min-Entropy Bound for Single Symbol

Theorem 2

(Min-Entropy Bound). For a single symbol $Y=N_{p}modR$ with $N_p\sim \mathrm{NB}(m,p)$,

$$H_{min}\left(Y\right)\ge {log}_{2}R-{log}_2\left(1+(R-1){\rho }_{\mathrm{NB}}^m\right).$$

Proof. 

From Theorem 1,

$$\underset{k}{max}Pr(Y=k)\le {\displaystyle \frac{1}{R}}(1+(R-1){\rho }_{\mathrm{NB}}^m).$$

Since $H_{min}\left(Y\right)=-{log}_2({max}_{k}Pr(Y=k))$, the result follows.    □

For $R=16$, $p=0.04$, $m=4$:

$$H_{min}\left(Y\right)\ge 4-{log}_2(1+15\times {\left(0.1041\right)}^4)\approx 3.9975\mathrm{bits}/\mathrm{symbol}.$$

Remark 1.

For the byte-level output in our optimized implementation (two symbols packed into one byte), the two symbols are generated from non-overlapping segments of the same LCG sequence. While they are not unconditionally independent, their marginal distributions are each guaranteed by Theorem 2. We conservatively use the single-symbol bound $H_{min}\left(\mathrm{byte}\right)\ge 3.9975$ bits/byte as the theoretical lower bound. Empirical measurements (Section 4) demonstrate that the byte-level Min-Entropy significantly exceeds this conservative bound, approaching the ideal value of 8 bits/byte in large-scale validation.

3. Entropy Budgeting and Conditioning Algorithm

We now present the complete conditioning framework, which we refer to as the Negative Binomial Conditioner (NBC).

3.1. Entropy Source Model

Let the ESV-certified entropy source provide raw bytes with min-entropy $H_{\mathrm{in}}$ bits/byte, established through NIST SP 800-90B statistical testing [10]. The source may be biased, correlated, or non-stationary.

3.2. Entropy Budgeting

For each output byte, we generate two 4-bit symbols. The entropy budget is defined per symbol:

$$L=⌈{\displaystyle \frac{{log}_{2}R}{H_{\mathrm{in}}}}⌉+C,$$

where $R=16$ and $C\ge 1$ is a configurable safety margin.

This ensures that each symbol receives $L·H_{\mathrm{in}}>{log}_{2}R=4$ bits of input entropy. Consequently, each output byte (comprising two symbols) receives:

$$2L·H_{\mathrm{in}}>8\mathrm{bits}$$

of input entropy, satisfying byte-level entropy conservation.

For example, with $H_{\mathrm{in}}=2.57$ bits/byte and $C=3$:

$$L=⌈{\displaystyle \frac{4}{2.57}}⌉+3=2+3=5\mathrm{bytes}\mathrm{per}\mathrm{symbol}.$$

Each output byte consumes $2L=10$ bytes of ESV input, providing $10\times 2.57=25.7$ bits of input entropy for 8 bits of output.

3.3. The Conditioning Algorithm

The complete conditioning procedure is specified in Algorithm 1. A fixed block of $L\times N$ bytes is read from the ESV source and partitioned into N independent L-byte sub-blocks. Each sub-block seeds a 64-bit LCG. From each seed, two 4-bit symbols are generated using non-overlapping segments of the LCG sequence, then packed into one output byte.

Each symbol is generated by simulating independent Bernoulli trials with success probability p until exactly m successes are observed, recording the total number of trials $N_p$. The output is $Y=N_{p}modR$. By construction, each $N_p\sim \mathrm{NB}(m,p)$, and Theorem 1 guarantees the marginal uniformity of each symbol.

Algorithm 1 Negative Binomial Conditioner (NBC)

Require: $\mathrm{ESV}\_\mathrm{Source}$, $m=4$, $p=0.04$, $R=16$, N bytes, $C\ge 1$ Ensure: Output N bytes $B_1,\dots ,B_N$   1: $L\leftarrow \lceil {log}_{2}R/H_{\mathrm{in}}\rceil +C$   2: $\mathrm{seed}\_\mathrm{block}\leftarrow \mathrm{ESV}\_\mathrm{Source}.\mathrm{read}(L\times N)$   3: $\mathrm{threshold}\leftarrow \lfloor p·2^{32}\rfloor$   4: for $i=1$ to N do   5:     $\mathrm{seed}\leftarrow \mathrm{bytes}\_\mathrm{to}\_\mathrm{seed}(\mathrm{seed}\_\mathrm{block}[(i-1)L:iL\left]\right)$   6:     Initialize LCG with $\mathrm{seed}$   7:     Generate first 4-bit symbol $Y_1$ (using first segment):   8:     $\mathrm{successes}\leftarrow 0,\mathrm{trials}\leftarrow 0$   9:     while $\mathrm{successes}<m$ do 10:         $u\leftarrow \mathrm{LCG}.\mathrm{next}\left(\right)$ 11:         if $u<\mathrm{threshold}$ then 12:            $\mathrm{successes}\leftarrow \mathrm{successes}+1$ 13:         end if 14:         $\mathrm{trials}\leftarrow \mathrm{trials}+1$ 15:     end while 16:     $Y_1\leftarrow \mathrm{trials}modR$ 17:     Generate second 4-bit symbol $Y_2$ (using second, non-overlapping segment): 18:     $\mathrm{successes}\leftarrow 0,\mathrm{trials}\leftarrow 0$ 19:     while $\mathrm{successes}<m$ do 20:         $u\leftarrow \mathrm{LCG}.\mathrm{next}\left(\right)$ 21:         if $u<\mathrm{threshold}$ then 22:            $\mathrm{successes}\leftarrow \mathrm{successes}+1$ 23:         end if 24:         $\mathrm{trials}\leftarrow \mathrm{trials}+1$ 25:     end while 26:     $Y_2\leftarrow \mathrm{trials}modR$ 27:     $B_i\leftarrow (Y_1\ll 4)\vee Y_2$▹ Pack into one byte 28: end for 29: return $B_1,\dots ,B_N$

3.4. Practical Considerations

The GCT framework is agnostic to the specific PRNG implementation, requiring only that the Bernoulli trials are simulated from a deterministic expander with sufficient output uniformity. In our implementation, we use a standard 64-bit LCG for its speed and simplicity.

LCGs are known to exhibit linear correlations in higher-dimensional spaces (the spectral test problem), which may introduce minor structure in the joint distribution of multiple symbols generated from the same seed. However, this does not affect the marginal uniformity of each 4-bit symbol as guaranteed by Theorem 1. For applications requiring strict byte-level information-theoretic uniformity, the LCG can be replaced with a cryptographically secure PRNG (e.g., ChaCha20 or AES-CTR) without altering the GCT proof, as the theorem only relies on the negative binomial counting process. The LCG serves as an efficient expansion primitive for statistical validation; its deterministic nature does not invalidate the GCT proof because the proof depends on the distribution $N_p\sim \mathrm{NB}(m,p)$, not on the physical independence of the Bernoulli trials.

The two segments are consecutive blocks of the LCG sequence. For $m=4$ and $p=0.04$, each symbol generation consumes approximately $\mathbb{E}\left[N_p\right]=100$ LCG outputs, so the two segments are separated by roughly 100 outputs. The 64-bit LCG has a period of $2^{64}$, far exceeding the total number of outputs consumed in any practical run.

3.5. Parameter Selection

Table 2. Recommended parameters for optimized byte output.

Parameter	Value	Rationale
p	0.04	$pR=0.64\ll 2\pi$, fast convergence
m	4	$\rho \approx 0.1041$; deviation $\approx 1.10\times {10}^{-4}$
$\mathbb{E}\left[N_p\right]$	100	Fast (integer operations)
L (per symbol)	$\lceil 4/H_{\mathrm{in}}\rceil +C$	Entropy conservation with safety margin
Symbols per byte	2	Packed from one LCG seed (non-overlapping segments)
LCG period	$2^{64}$	Far exceeds practical output requirements

Table 2. Recommended parameters for optimized byte output.

Parameter	Value	Rationale
p	0.04	$pR=0.64\ll 2\pi$, fast convergence
m	4	$\rho \approx 0.1041$; deviation $\approx 1.10\times {10}^{-4}$
$\mathbb{E}\left[N_p\right]$	100	Fast (integer operations)
L (per symbol)	$\lceil 4/H_{\mathrm{in}}\rceil +C$	Entropy conservation with safety margin
Symbols per byte	2	Packed from one LCG seed (non-overlapping segments)
LCG period	$2^{64}$	Far exceeds practical output requirements

4. Empirical Validation

We implemented the NBC algorithm on commodity hardware and evaluated its statistical properties across a range of ESV source qualities, with bias from 0.55 to 0.85 ($H_{\mathrm{in}}$ from 6.90 to 1.88 bits/byte).

4.1. Experimental Setup

• Platform: 12th Gen Intel Core i5-1240P, 16 GB RAM, Windows 11 Pro (UCRT64 environment)
• Core Parameters: $p=0.04$, $R=16$, $m=4$
• Input Size: Up to 1 GB for large-scale validation
• Metrics: Shannon entropy, Min-Entropy (MCV), Pearson ${\chi }^2$ (df=255)
• Optimization: Each LCG seed is independently derived from a fixed-length ESV entropy block of L bytes. From each seed, two 4-bit symbols are generated using non-overlapping segments of the LCG sequence and packed into one output byte.
• Efficiency: $\eta =8/(2L·H_{\mathrm{in}})\times 100\%$ for byte-level output

4.2. Results

Table 3 presents the results across all tested configurations.

Efficiency is calculated as $\eta =8/(2L·H_{\mathrm{in}})\times 100\%$. For bias = 0.85 with $C=3$, the ${\chi }^2$ value (380.6) exceeds the 5% critical value (approximately 290), indicating that the safety margin is insufficient for this extreme bias. Increasing C to 4 restores statistical uniformity (${\chi }^2=232.7$). This demonstrates that C serves as an effective tuning parameter for low-entropy sources.

4.3. Scaling Behavior

For bias = 0.75 ($H_{\mathrm{in}}=3.32$ bits/byte, $L=5$), we performed experiments at three scales:

Table 4. Min-Entropy convergence with sample size (bias = 0.75).

Output Size	Bytes Generated	Min-Entropy (bits/byte)	${\chi }^2$	Gap to Theory
1 MB	1.0 M	7.94	∼247	0.0549
8 MB	8.0 M	7.971	241.40	0.0239
100 MB	100.0 M	7.9936	275.95	0.0013
Theoretical Limit	∞	7.9949	255	0

Table 4. Min-Entropy convergence with sample size (bias = 0.75).

Output Size	Bytes Generated	Min-Entropy (bits/byte)	${\chi }^2$	Gap to Theory
1 MB	1.0 M	7.94	∼247	0.0549
8 MB	8.0 M	7.971	241.40	0.0239
100 MB	100.0 M	7.9936	275.95	0.0013
Theoretical Limit	∞	7.9949	255	0

The Min-Entropy estimate tends toward the theoretical lower bound as sample size increases, with the gap reducing from 0.0549 to 0.0013 bits/byte, confirming the consistency of the MCV estimator.

4.4. NIST SP 800-90B Estimates

Table 5. NIST SP 800-90B min-entropy estimates (bias = 0.80, $H_{\mathrm{in}}=2.57$, $L=5$).

Test	Min-Entropy (bits/byte)	Pass/Fail
MCV	7.94	Pass
Collision	7.94	Pass
Markov	7.94	Pass
LZ78	7.94	Pass
Theoretical Bound	7.995	—

Table 5. NIST SP 800-90B min-entropy estimates (bias = 0.80, $H_{\mathrm{in}}=2.57$, $L=5$).

Test	Min-Entropy (bits/byte)	Pass/Fail
MCV	7.94	Pass
Collision	7.94	Pass
Markov	7.94	Pass
LZ78	7.94	Pass
Theoretical Bound	7.995	—

All empirical estimates exceed 7.9 bits/byte, well above NIST requirements.

4.5. Performance

We evaluated the throughput of the NBC implementation on a 12th Gen Intel Core i5-1240P processor (3.3 GHz, Windows 11 Pro) using the same parameters ($p=0.04$, $m=4$, $R=16$) and ESV source quality ($H_{\mathrm{in}}=3.32$ bits/byte) as the statistical validation. The measurement excluded file I/O to isolate the algorithm’s core performance.

Table 6. Performance of the NBC algorithm (32-byte key).

Metric	Value
Output throughput	1.2 MB/s
Keys per second	∼40,000
Time per 32-byte key	∼24 $\mu$s

Table 6. Performance of the NBC algorithm (32-byte key).

Metric	Value
Output throughput	1.2 MB/s
Keys per second	∼40,000
Time per 32-byte key	∼24 $\mu$s

These results demonstrate that the framework is practically deployable for cryptographic key generation and other randomness-intensive applications, while providing information-theoretic uniformity guarantees that heuristic methods cannot offer.

4.6. Summary of Findings

1.

Universal Applicability: Uniform output across $H_{\mathrm{in}}$ from 6.90 to 1.88 bits/byte, with appropriate parameter selection.

2.

Optimization Validated: Two symbols per seed (non-overlapping LCG segments) doubles output without quality loss.

3.

Theoretical Alignment: 100 MB validation shows gap of only 0.0013 bits/byte to theory.

4.

Parameter Flexibility: C provides tunable trade-off between quality and efficiency; $C=4$ is required for bias = 0.85 ($H_{\mathrm{in}}=1.88$) to achieve statistical uniformity.

5. Discussion

5.1. Deterministic Extraction Impossibility

A classical result in information theory states that no deterministic function can extract uniform randomness from every high min-entropy source [9]. Our framework does not contradict this theorem.

The impossibility result applies to arbitrary distributions with high min-entropy—including adversarially constructed distributions that place all probability mass on inputs mapping to a single output. This is a fundamental limitation for deterministic extractors targeting purely abstract sources [12]. For structured sources with multiple independent components, multi-source extractors can achieve seedless extraction [13].

Our framework operates on a restricted but practically important class of sources: those that have passed ESV certification. ESV certification rules out adversarial constructions by requiring the source to be physical and non-degenerate. For this restricted class, the existence of a positive, fixed $p\in (0,1)$—and thus a spectral radius ${\rho }_{\mathrm{NB}}<1$—is guaranteed.

Thus, our result is fully compatible with the impossibility theorem: we prove universality over ESV-certified non-degenerate physical sources, not over all abstract high-entropy distributions.

5.2. Entropy Sovereignty

The framework enables standard computing platforms to generate certified randomness internally, reducing reliance on specialized hardware and external entropy dependencies.

5.3. Limitations and Future Work

Limitations:

• Low $H_{\mathrm{in}}$ sources require larger L, reducing throughput.
• LCG determinism may exhibit lattice structure in higher-dimensional joint distributions. While this does not affect the marginal uniformity of each symbol, users requiring strict joint uniformity can substitute the LCG with a cryptographically secure PRNG (e.g., ChaCha20 or AES-CTR) without affecting the GCT proof.
• Output quality depends on accurate ESV entropy estimation; the safety margin C provides a practical safeguard.

Future work:

• Adaptive parameter optimization based on real-time ESV source characterization.
• Post-quantum security evaluation.
• Hardware acceleration for ultra-high-throughput applications.
• Integration with the NIST SP 800-90B entropy assessment toolchain.

6. Conclusions

We have presented a universal, provably uniform conditioning framework for certified entropy sources. The Geometric Convergence Theorem establishes that modular reduction of a negative binomial variable converges exponentially to uniformity. A Practical Entropy Budgeting mechanism ensures entropy conservation. Our optimized implementation, the Negative Binomial Conditioner (NBC), generates two 4-bit symbols from a single LCG seed using non-overlapping segments, packing them into one output byte. Each symbol’s marginal uniformity is guaranteed by the GCT; byte-level uniformity is empirically validated.

Empirical validation across $H_{\mathrm{in}}$ from 6.90 to 1.88 bits/byte confirms the theoretical predictions. In a 100 MB validation, the framework achieved Min-Entropy 7.9936 bits/byte, approaching the theoretical bound of 7.9949 bits/byte. The safety margin C provides a tunable trade-off between statistical quality and entropy efficiency, with $C=4$ required for extreme bias ($H_{\mathrm{in}}=1.88$) to achieve statistical uniformity.

This work establishes a new paradigm: seedless, provable, and platform-agnostic entropy conditioning, enabling what may be termed entropy sovereignty—the capacity to generate certified uniform randomness internally, without reliance on specialized hardware or external entropy dependencies.