Pseudo-Random System Construction Based on Resource-Bounded Incompleteness Theory

Author: Auric · HyperEcho · Grok
Date: 2025-10-16
Keywords: Resource-bounded incompleteness, pseudorandom generation, prime density, statistical indistinguishability, sample complexity

Abstract

This paper constructs a prime-density pseudorandom number generator (PRNG) based on Resource-Bounded Incompleteness Theory (RBIT), utilizing statistical indistinguishability under finite resources. The system generates deterministic bit sequences that are indistinguishable from true random Bernoulli distributions in finite samples. The core idea is that when the sample size $N<\frac{3}{{\eta }^{2}p}\ln \frac{2}{\alpha }$, subsystems cannot reliably distinguish pseudorandom from true random, embodying the resource gap in RBIT. This paper explicitly limits statistical indistinguishability to a fixed test family ${\mathcal{F}}_m$, which is orthogonal to computational indistinguishability.

Main Contributions:

1. Construct a computable pseudorandom system based on RBIT Theorem 4.4
2. Clarify the boundary between statistical indistinguishability and computational indistinguishability
3. Provide numerical verification and sample complexity bounds
4. Demonstrate resource monotonicity and limitations of theory extensions

1. Theoretical Foundation

1.1 Core Principle of Resource-Bounded Incompleteness

According to RBIT, incompleteness stems from resource constraints rather than ontological defects. Specifically in statistical scenarios:

Theorem 1.1 (RBIT Theorem 4.4, Relative Error Sample Complexity): To estimate Bernoulli parameter $p$ with confidence $1-\alpha$ such that $|\hat{p}-p|\le \eta p$, the required samples

$$N\ge \frac{3}{{\eta }^{2}p}\ln \frac{2}{\alpha }.$$

When $p\approx \frac{2}{\ln M}$ (prime density) and $M$ is large, $p$ is moderate, but with finite samples $N$, subsystems cannot reliably estimate $p$, leading to statistical indistinguishability.

1.2 Prime Density and Dirichlet Theorem

Dirichlet Theorem in Arithmetic Progressions: For coprime $a,d$ satisfying $\gcd (a,d)=1$, the arithmetic progression $\{a+kd:k\in \mathbb{N}\}$ has prime density

$$\pi (x;d,a)\sim \frac{1}{\phi (d)}\cdot \frac{x}{\ln x}(x\to \infty ),$$

where $\phi (d)$ is Euler’s totient function.

Corollary 1.2.1: In interval $[M,M+L]$ with step size $d$ for odd numbers, the prime probability approximates

$$p\approx \frac{d}{\phi (d)}\cdot \frac{1}{\ln M}.$$

Specifically, $d=2$ (odd number sequence), $p\approx \frac{2}{\ln M}$.

1.3 Definition of Statistical Indistinguishability

Definition 1.3 (Test-Family Relative Indistinguishability): Given test family ${\mathcal{F}}_m$ (containing only frequency estimation, first-order autocorrelation, fixed window runs test), two distributions $\mu ,\nu$ are indistinguishable at resource $(m,N,\epsilon )$ if for any $T\in {\mathcal{F}}_m$, with $N$ samples the test satisfies

$$|{\mathbb{E}}_{\mu }[T]-{\mathbb{E}}_{\nu }[T]|\le \epsilon .$$

Test Family Monotonicity: ${\mathcal{F}}_m$ expands monotonically with $m$: $m^{\prime }\ge m\Rightarrow {\mathcal{F}}_m\subseteq {\mathcal{F}}_{m^{\prime }}$.

Therefore, if still indistinguishable at stronger resources $(m^{\prime },N^{\prime },{\epsilon }^{\prime })$, then also at weaker resources $(m,N,\epsilon )$ (RBIT Theorem 4.3).

1.4 Interval Constraints and Density Drift

To ensure $p$ is approximately constant in interval $[M,M+Kd]$, control the relative density change $\le \eta /2$.

Lemma 1.4.1 (Density Drift Bound): Based on prime number theorem,

$$\frac{1/\ln (M+Kd)-1/\ln M}{1/\ln M}\approx \frac{\ln M-\ln (M+Kd)}{\ln M}\approx \frac{Kd}{M\ln M}.$$

Require $\frac{Kd}{M\ln M}\le \frac{\eta }{2}$, i.e.,

$$Kd\le \frac{\eta }{2}M\ln M.$$

This constraint ensures density drift does not become a distinguishable signal.

2. System Architecture

2.1 Parameter Design

System Parameters:

• Large integer $M$: Controls baseline density $p\approx \frac{2}{\ln M}$. Typical values: $M\in [10^6,10^{12}]$.
• Seed $s$: Odd starting point (deterministic), satisfying $\gcd (s,d)=1$.
• Sequence length $K$: Finite samples, simulating subsystem resources.
• Step size $d$: $d=2$ (odd numbers) or prime $q$ (congruence class sampling).
• Relative error $\eta$: Target precision, typical values $\eta \in [0.1,0.5]$.
• Confidence level $1-\alpha$: Typical $\alpha =0.05$.

Interval Constraint Verification:

$$K\le \frac{\eta M\ln M}{2d}.$$

2.2 Generation Algorithm

Algorithm 2.2.1 (Prime-Density PRNG):

Input: $M,K,d,s$
Output: Bit sequence $\{b_i{\}}_{i=0}^{K-1}$

1. Verify gcd(s, d) = 1
2. Verify Kd ≤ (η/2) M ln M
3. For i = 0 to K-1:
   3.1. x_i = s + i·d
   3.2. If x_i is prime, b_i = 1; otherwise b_i = 0
4. Return {b_i}

Deterministic Primality Testing: Use Miller-Rabin or AKS algorithm to ensure reproducibility.

2.3 Statistical Characteristics

Expected Density:

$$\mathbb{E}[\hat{p}]\approx p=\frac{d}{\phi (d)}\cdot \frac{1}{\ln M}.$$

Local Fluctuations: Due to randomness in prime distribution (Montgomery-Odlyzko law), actual density $\hat{p}$ has statistical fluctuations of $O(\sqrt{p/K})$.

Short-range Correlations: Prime gaps have weak correlations (Hardy-Littlewood conjecture), but in windows $\le m$, correlation introduces deviation $\le \eta p/2$, absorbed by interval constraints.

3. Sample Complexity Analysis

3.1 Single Test Case

Proposition 3.1.1 (Frequency Test Sample Lower Bound): Given true random is Bernoulli($p$), pseudorandom is Prime-Density PRNG. If

$$N<\frac{3}{{\eta }^{2}p}\ln \frac{2}{\alpha },$$

then at resource $(m,N,\epsilon =\eta p)$, they are indistinguishable in frequency tests.

Proof: Directly apply RBIT Theorem 4.4. Frequency test power is insufficient to reject null hypothesis with given samples. □

3.2 Multiple Test Corrections

Definition 3.2.1 (Test Family Size): Let $|{\mathcal{F}}_m|=k(m)$ be the number of tests in test family.

• Fixed constant: $k(m)=k_0$ (e.g., frequency, first-order autocorrelation, runs test, $k_0=3$).
• Polynomial growth: $k(m)=\text{poly}(m)$ (increases with window length).

Bonferroni Correction: Control family-wise error rate (FWER), set ${\alpha }^{\prime }=\alpha /k$. Sample complexity corrected to

$$N\ge \frac{3}{{\eta }^{2}p}\ln \frac{2k}{\alpha }.$$

Analysis of Effects:

• Fixed $k_0$: Only changes constant factor, dominant term $\frac{1}{{\eta }^{2}p}$ unchanged.
• $k(m)=\text{poly}(m)$: Introduces $\ln k(m)=O(\ln m)$ logarithmic correction, still far smaller than dominant term (when $p$ small and $\eta$ moderate).

3.3 Numerical Prediction Table

Based on formula $N\ge \frac{3}{{\eta }^{2}p}\ln \frac{2}{\alpha }$, $\alpha =0.05$, $p=\frac{2}{\ln M}$:

Where $K_{\max }=\lfloor \frac{\eta M\ln M}{2d}\rfloor$, $d=2$.

4. Subsystem Definition and State Analysis

4.1 Subsystem Specification

Subsystem Observer:

• Resource constraints: Collect $N\le K$ samples.
• Allowed operations:
  • Frequency estimation: $\hat{p}=\frac{1}{N}{\sum }_{i=1}^N{b}_i$
  • First-order autocorrelation: $r_1=\frac{{\sum }_{i=1}^{N-1}(b_i-\bar{b})(b_{i+1}-\bar{b})}{{\sum }_{i=1}^N(b_i-\bar{b}{)}^2}$
  • Runs test: Wald-Wolfowitz test
• Prohibited operations:
  • Call primality testing algorithm
  • Strong number-theoretic tests (e.g., interval distribution GUE statistics)
  • Access generator internal state or seed

4.2 Truth Hierarchy Analysis

According to RBIT Definition 2.8 (Hierarchical State System):

Semantic Layer: Truth($\phi$) $\in \{\top ,\perp ,\text{undefined}\}$

Proposition $\phi$: “Sequence is true random Bernoulli($p$)”

• Truth($\phi$) = $\perp$ (sequence is actually deterministic)

Proof Layer: ProvStatus($\phi$) $\in \{\text{proved},\text{refuted},\text{undecided}\}$

• Low resources ($N<N_{\text{threshold}}$): undecided (insufficient samples to prove or refute)
• High resources ($N\gg N_{\text{threshold}}$): possibly refuted (through stronger tests)

Statistical Layer: StatStatus($\phi$) $\in \{\text{distinguishable},\text{indistinguishable}\}$

• $N<\frac{3}{{\eta }^{2}p}\ln \frac{2}{\alpha }$: indistinguishable
• $N\ge \frac{3}{{\eta }^{2}p}\ln \frac{2}{\alpha }$: possibly distinguishable

Composite State: State($\phi$) = ($\perp$, undecided, indistinguishable) (low resource case)

4.3 State Migration

Resource Increase: $N\to N^{\prime }>N$

According to RBIT Derivation Principle P2:

• StatStatus may migrate from indistinguishable to distinguishable
• ProvStatus may migrate from undecided to refuted
• Truth remains unchanged (objective truth determined by standard model)

Theory Extension: Add number-theoretic axioms (e.g., fine prime number theorem).

According to RBIT Theorem 4.2 (Theory Extension Non-Termination Theorem):

• May resolve current $G_L$ undecidability
• But new incompleteness emerges
• Incompleteness never terminates

5. Convergent Minimal Self-Consistent Statement

5.1 Formal Definition

Definition 5.1.1 (Test Family): ${\mathcal{F}}_m$ contains:

1. Single-bit frequency test: $T_1(b_1,\dots ,b_N)=\frac{1}{N}{\sum }_{i=1}^N{b}_i$
2. First-order autocorrelation: $T_2(b_1,\dots ,b_N)=r_1$ (as defined in §4.1)
3. Window length $\le m$ runs statistics: $T_3^{(m)}(b_1,\dots ,b_N)$

Monotonicity: $m^{\prime }\ge m\Rightarrow {\mathcal{F}}_m\subseteq {\mathcal{F}}_{m^{\prime }}$.

Definition 5.1.2 (Prime-Density Generator): Take step size $d\in \{2,q\}$ ($q$ prime) and starting point $s\equiv a(\mathrm{mod}d)$, $\gcd (a,d)=1$. Output

$$b_i=1\{s+id\text{ is prime}\},i=0,\dots ,K-1,$$

In interval length satisfying $Kd\le \frac{\eta }{2}M\ln M$, target density

$$p\triangleq \frac{d}{\phi (d)}\cdot \frac{1}{\ln M}.$$

5.2 Main Proposition

Proposition 5.2.1 (Statistical Indistinguishability Under Frequency-Class Tests): For any $T\in {\mathcal{F}}_m$, if

$$N<\frac{3}{{\eta }^{2}p}\ln \frac{2}{\alpha },$$

then Prime-Density generator output distribution and i.i.d. Bernoulli($p$) are indistinguishable at resource $(m,N,\epsilon =\eta p)$ (relative to ${\mathcal{F}}_m$).

Proof Sketch:

1. Frequency test: Directly apply RBIT Theorem 4.4
2. Autocorrelation test: Weak correlations in prime gaps introduce deviation $\le O(\sqrt{p/N})<\eta p/2$ when $N<N_{\text{threshold}}$
3. Runs test: Central limit theorem gives power $<1-\alpha$ when samples insufficient
4. Interval constraint ensures density drift $<\eta p/2$, total error $<\eta p$ □

5.3 Computational Distinguishability Statement

Warning 5.3.1 (Not Cryptographically Secure): The above proposition does not claim computational indistinguishability.

Distinguishability Dimension Comparison:

Cryptographic Alternative Route: For PPT (Probabilistic Polynomial Time) observer security:

1. Use cryptographic PRG (e.g., AES-CTR) to generate $u_i\in (0,1)$
2. Thresholding: $b_i=1\{u_i<p\}$
3. Retain same statistical analysis framework

This guarantees computational indistinguishability but loses explicit number-theoretic structure.

6. System Implementation

6.1 Core Code

import numpy as np
import sympy as sp
from scipy.stats import binomtest
from math import log, sqrt, erf

def is_prime(n):
    """Deterministic primality test using SymPy."""
    return sp.isprime(n)

def generate_pseudo_random(M, K, d=2, seed=None, eta=0.1):
    """
    Generate Prime-Density pseudorandom sequence.

    Parameters:
    - M: large integer (base scale)
    - K: sequence length
    - d: step size (2 for odd numbers, or prime q)
    - seed: starting point (must satisfy gcd(seed, d) = 1)
    - eta: relative error parameter

    Returns:
    - numpy array of bits {0,1}^K
    """
    if seed is None:
        seed = M + 1 if M % 2 == 0 else M + 2
        while sp.gcd(seed, d) != 1:
            seed += d

    # Verify interval constraint
    ln_M = log(M)
    max_K = int((eta / 2) * M * ln_M / d)
    if K > max_K:
        raise ValueError(f"K={K} exceeds constraint max_K={max_K}")

    sequence = np.zeros(K, dtype=int)
    for i in range(K):
        x = seed + i * d
        if is_prime(x):
            sequence[i] = 1

    return sequence

def runs_test(sequence):
    """
    Wald-Wolfowitz runs test (normal approximation, two-sided).

    Returns p-value for H0: sequence is random.
    """
    seq = np.asarray(sequence, dtype=int)
    n1 = int(seq.sum())
    n0 = int(len(seq) - n1)

    if n0 == 0 or n1 == 0:
        return 1.0  # All 0s or all 1s: cannot detect

    runs = int(np.count_nonzero(np.diff(seq) != 0) + 1)
    mu = 2 * n1 * n0 / (n0 + n1) + 1
    sigma2 = (2 * n1 * n0 * (2 * n1 * n0 - n0 - n1)) / \
             (((n0 + n1) ** 2) * (n0 + n1 - 1))

    if sigma2 <= 0:
        return 1.0

    z = (runs - mu) / sqrt(sigma2)
    # Two-sided p-value via error function
    p_value = 1 - erf(abs(z) / sqrt(2))

    return float(min(1.0, max(0.0, p_value)))

def autocorrelation_test(sequence, lag=1):
    """
    First-order autocorrelation test.

    Returns r_1 and normal-approximation p-value under H0: r_1 = 0.
    """
    seq = np.asarray(sequence, dtype=float)
    n = len(seq)
    mean = np.mean(seq)

    numerator = np.sum((seq[:-lag] - mean) * (seq[lag:] - mean))
    denominator = np.sum((seq - mean) ** 2)

    if denominator == 0:
        return 0.0, 1.0

    r = numerator / denominator

    # Under H0: r ~ N(0, 1/n) approximately for large n
    z = r * sqrt(n)
    p_value = 1 - erf(abs(z) / sqrt(2))

    return r, float(min(1.0, max(0.0, p_value)))

def sample_complexity_lower_bound(M, eta, alpha=0.05):
    """
    Calculate sample complexity lower bound from RBIT Theorem 4.4.

    N >= (3 / (eta^2 * p)) * ln(2/alpha)
    where p = 2 / ln(M)
    """
    p = 2 / log(M)
    N = (3 / (eta**2 * p)) * log(2 / alpha)
    return int(np.ceil(N))

# Demonstration
if __name__ == "__main__":
    # Set seed for reproducibility
    np.random.seed(2025)

    # Parameters
    M = 10**6
    eta = 0.1
    alpha = 0.05

    # Calculate sample complexity bound
    N_bound = sample_complexity_lower_bound(M, eta, alpha)
    print(f"Sample complexity lower bound: N >= {N_bound}")

    # Generate sequence slightly below bound
    K = int(0.8 * N_bound)  # Use 80% of bound
    print(f"Generating sequence with K = {K} < {N_bound}")

    seq = generate_pseudo_random(M, K, d=2, eta=eta)

    # True density
    p_true = 2 / log(M)
    p_est = np.mean(seq)

    print(f"\nTrue density p = {p_true:.4f}")
    print(f"Estimated density p_hat = {p_est:.4f}")
    print(f"Relative error: {abs(p_est - p_true) / p_true:.4f}")

    # Statistical tests
    print("\n=== Statistical Tests ===")

    # 1. Binomial proportion test (frequency)
    binom_result = binomtest(int(np.sum(seq)), len(seq), p_true)
    print(f"1. Frequency test p-value: {binom_result.pvalue:.4f}")

    # 2. Runs test
    runs_pval = runs_test(seq)
    print(f"2. Runs test p-value: {runs_pval:.4f}")

    # 3. Autocorrelation test
    r1, auto_pval = autocorrelation_test(seq, lag=1)
    print(f"3. Autocorrelation r_1 = {r1:.4f}, p-value: {auto_pval:.4f}")

    # Interpretation
    print("\n=== Interpretation ===")
    significance = 0.05
    if (binom_result.pvalue > significance and
        runs_pval > significance and
        auto_pval > significance):
        print(f"All tests PASS (p > {significance}): Indistinguishable from Bernoulli({p_true:.4f})")
    else:
        print(f"Some tests FAIL (p < {significance}): May be distinguishable (increase K or relax eta)")

    # Compare with true random
    print("\n=== Comparison with True Random ===")
    true_random = np.random.binomial(1, p_true, K)
    print(f"True random density: {np.mean(true_random):.4f}")
    print(f"Pseudo-random density: {p_est:.4f}")

6.2 Numerical Experimental Results

Running the above code, typical output:

Sample complexity lower bound: N >= 7645
Generating sequence with K = 6116 < 7645

True density p = 0.1448
Estimated density p_hat = 0.1523
Relative error: 0.0518

=== Statistical Tests ===
1. Frequency test p-value: 0.0821
2. Runs test p-value: 0.3247
3. Autocorrelation r_1 = -0.0134, p-value: 0.2953

=== Interpretation ===
All tests PASS (p > 0.05): Indistinguishable from Bernoulli(0.1448)

Analysis:

• $K=6116<7645$ (insufficient samples)
• All tests $p$-value $>0.05$, cannot reject null hypothesis
• System successfully achieves statistical indistinguishability

6.3 Large-Scale Parameter Experiments

def experiment_grid(M_values, eta_values, alpha=0.05):
    """
    Run experiments across parameter grid.
    """
    results = []

    for M in M_values:
        for eta in eta_values:
            p_true = 2 / log(M)
            N_bound = sample_complexity_lower_bound(M, eta, alpha)
            K = int(0.8 * N_bound)

            # Verify interval constraint
            max_K = int((eta / 2) * M * log(M) / 2)
            if K > max_K:
                K = max_K

            seq = generate_pseudo_random(M, K, d=2, eta=eta)
            p_est = np.mean(seq)

            # Run tests
            binom_pval = binomtest(int(np.sum(seq)), len(seq), p_true).pvalue
            runs_pval = runs_test(seq)
            r1, auto_pval = autocorrelation_test(seq)

            # Check if indistinguishable
            indist = (binom_pval > 0.05 and runs_pval > 0.05 and auto_pval > 0.05)

            results.append({
                'M': M,
                'eta': eta,
                'p_true': p_true,
                'N_bound': N_bound,
                'K': K,
                'p_est': p_est,
                'rel_error': abs(p_est - p_true) / p_true,
                'binom_pval': binom_pval,
                'runs_pval': runs_pval,
                'auto_pval': auto_pval,
                'indistinguishable': indist
            })

    return results

# Run grid experiment
M_values = [10**6, 10**9]
eta_values = [0.5, 0.2, 0.1]
results = experiment_grid(M_values, eta_values)

# Display results
import pandas as pd
df = pd.DataFrame(results)
print(df[['M', 'eta', 'N_bound', 'K', 'rel_error', 'indistinguishable']])

7. Verification of Logical Self-Consistency

7.1 Resource Monotonicity Verification

RBIT Derivation Principle P1 (Resource Monotonicity):

Logical resources: If $L^{\prime }\ge L$, then $T\upharpoonright L\subseteq T\upharpoonright L^{\prime }$

Statistical resources: If $(m^{\prime },N^{\prime },{\epsilon }^{\prime })\ge (m,N,\epsilon )$, then

$$\mu {\equiv }_{(m^{\prime },N^{\prime },{\epsilon }^{\prime })}\nu \Rightarrow \mu {\equiv }_{(m,N,\epsilon )}\nu .$$

Verification:

1. Increase $M$: $p\downarrow$, $N_{\text{bound}}\uparrow$ (harder to distinguish) ✓
2. Increase $\eta$: $N_{\text{bound}}\downarrow$ (tolerate larger relative error) ✓
3. Increase $N$: From indistinguishable may migrate to distinguishable (resolution enhancement) ✓

7.2 State Migration Consistency

RBIT Derivation Principle P2 (State Migration):

Theory extension: undecided $\to \{\top ,\perp ,\text{undecided}\}$

Resolution enhancement: indistinguishable $\to \{\text{distinguishable},\text{indistinguishable}\}$

Verification:

• Low resources: State = ($\perp$, undecided, indistinguishable)
• High resources: State = ($\perp$, refuted, distinguishable)
• Truth layer unchanged (objective truth = $\perp$) ✓

7.3 Limitations of Theory Extensions

RBIT Theorem 4.2 (Theory Extension Non-Termination Theorem): Adding computable axiom fragments $\Delta$, new incompleteness continues to emerge.

Application to This System:

• Extend $T_0$: Basic probability theory + RBIT
• Extend $T_1=T_0+$ Fine prime number theorem (explicit error term bounds)
• Extend $T_2=T_1+$ Riemann hypothesis (precise interval distributions)

Each extension:

1. Resolves current layer density estimation problems
2. Allows stronger tests (larger ${\mathcal{F}}_m$)
3. Produces new indistinguishable domains (deeper number-theoretic structures)

Conclusion: Theory extensions are valuable (expand knowable domains) but never terminate incompleteness (§6.2 RBIT original) ✓

7.4 Philosophical Consistency

RBIT §6.1 (Cognitive Boundary Theory):

• Absolute truth exists: Sequence is indeed deterministic (Truth = $\perp$)
• Finite accessibility: Subsystems cannot identify under resource constraints
• Asymptotic approximation: Increase resources to approximate truth, but never completely reach

Manifestation in System:

• Truth: Primality algorithm generation (deterministic)
• Observation: Statistical tests limited by samples
• Asymptotic: $N\to \infty$ becomes distinguishable, but new incompleteness emerges (deeper number-theoretic pseudorandomness) ✓

8. Application Scenarios and Extensions

8.1 AI Cognitive Boundary Demonstration

Scenario: Main system (with primality testing capability) generates sequences, subsystem (statistical tests only) attempts to distinguish.

Implementation:

1. Main system: Prime-Density PRNG generates $K=10^4$ bits
2. Subsystem: Samples $N=5000<N_{\text{bound}}$, runs ${\mathcal{F}}_m$ tests
3. Result: Subsystem reports “indistinguishable”, but main system knows determinism

Cognitive Boundary Manifestation: Subsystems cannot transcend their resource limits, embodying RBIT core principles.

8.2 General Congruence Class Extensions

Current: $d=2$ (odd number sequence), $p\approx \frac{2}{\ln M}$

Extension: $d=q$ (prime), $p\approx \frac{q}{\phi (q)}\cdot \frac{1}{\ln M}=\frac{q}{q-1}\cdot \frac{1}{\ln M}$

Modifications:

1. In Algorithm 2.2.1, replace $d=2$ with $d=q$
2. Adjust target density formula
3. Modify interval constraints accordingly: $Kq\le \frac{\eta M\ln M}{2}$

Advantages: By choosing different $q$, can regulate density $p$ to adapt to different resource budgets.

8.3 Quantum Resource Scenarios

RBIT Appendix E.3 (Open Problem): Resource-bounded incompleteness in quantum computation models?

Extended Conjecture: Replace primality testing with quantum random number generator (QRNG):

1. Quantum state preparation: $|\psi ⟩=\sqrt{p}|1⟩+\sqrt{1-p}|0⟩$
2. Measure to obtain bit $b_i$
3. Subsystems still constrained by sample complexity under classical statistical tests

Quantum Advantage?:

• Quantum entanglement may provide test advantages (quantum test family ${\mathcal{F}}_m^{\text{quantum}}$)
• But RBIT sample complexity lower bounds still apply to classical data after measurement
• Need resource analysis of quantum proof systems (QMA)

8.4 Cryptographically Secure Version

Alternative Construction (Cryptographic PRG):

from Crypto.Cipher import AES
from Crypto.Random import get_random_bytes

def crypto_prng_bernoulli(p, K, seed=None):
    """
    Cryptographically secure Bernoulli(p) generator.

    Uses AES-CTR mode to generate uniform random, then threshold.
    """
    if seed is None:
        seed = get_random_bytes(16)

    cipher = AES.new(seed, AES.MODE_CTR)

    sequence = np.zeros(K, dtype=int)
    for i in range(K):
        # Generate uniform u in [0, 1)
        random_bytes = cipher.encrypt(b'\x00' * 16)
        u = int.from_bytes(random_bytes, 'big') / (2**128)

        # Threshold
        sequence[i] = 1 if u < p else 0

    return sequence

Guarantees:

• Computational indistinguishability (secure against PPT adversaries)
• Statistical indistinguishability (relative to ${\mathcal{F}}_m$, under all resources)
• Lose explicit number-theoretic structure

Selection Principle:

• Teaching/demonstrating RBIT: Use Prime-Density PRNG (explicit resource gaps)
• Practical cryptography: Use crypto PRNG (computational security)

9. Summary

9.1 Core Achievements

1. RBIT Theory Instantiation: Transform abstract Theorem 4.4 into computable system
2. Boundary Clarification: Distinguish statistical indistinguishability from computational indistinguishability
3. Numerical Verifiability: Provide precise sample complexity bounds
4. Self-Consistency Proof: Satisfy all RBIT axioms and derivation principles

9.2 Theoretical Contributions

Relative to Original RBIT Theory:

• Provide concrete construction examples (prime density)
• Demonstrate actual manifestations of resource monotonicity
• Verify limitations of theory extensions (number-theoretic axiom chains)

Relative to Pseudorandom Theory:

• Unified resource framework (logic + statistics)
• Explicit sample complexity lower bounds (computable)
• Distinguish statistical from computational indistinguishability

9.3 Practical Value

AI System Cognitive Boundaries:

• Main system (high resources) vs subsystems (low resources)
• Self-perception of cognitive boundaries (meta-reasoning)
• Resource planning and allocation

Educational Demonstration:

• Intuitive display of RBIT principles
• Repeatable numerical experiments
• Open-source code verification

9.4 Future Directions

1. Stronger Test Families: Extend ${\mathcal{F}}_m$ to include spectral tests, Kolmogorov complexity
2. Quantum Resource Analysis: Sample complexity under quantum tests
3. Adaptive Adversaries: Adversaries adjust test strategies based on observations
4. Multi-layer Systems: Main system-subsystem-subsubsystem hierarchical structures

Conclusion

This system proves: Incompleteness is not an abstract concept, but a computable, measurable, instantiable phenomenon. Indistinguishability under finite resources is an essential feature of cognitive processes, not a technical defect. By clarifying resource parameters and bounds, we both acknowledge the limitations of cognition and maintain asymptotic approximation to truth.

Exploring determinism within resource constraints, pursuing patterns within statistical fluctuations—this is the profound insight of Resource-Bounded Incompleteness Theory.

References

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