Finite-Precision Safe-Zone Extraction and Lyapunov Scaling in Coupled Chaotic Oscillators for Pseudorandom Number Generation

Finite-precision arithmetic can alter the statistical properties of chaotic trajectories, making bit extraction a substantive part of chaos-based pseudorandom number generation. We study MCL (Madeeh Chaotic Lock), a coupled phase-oscillator system with asymmetric coprime coupling (p = 3, q = 5), rationally independent irrational frequencies, and Gauss-Seidel sequential update. Position-resolved Chi-Square tests identify a reproducible byte-extraction region, the Safe Zone, in its XOR-mixed IEEE 754 binary64 mantissas: all 34 eight-bit windows starting at positions [6, 39] pass the byte-level uniformity test at 10⁸ samples in every seed tested, between a rounding-contaminated low band and an exponent-correlated high band. Controls reproduce the three-band structure in the Chirikov standard and Hénon maps; the logistic map shows how finite-precision cycles confound such scans. A dual-window extractor passes BigCrush with zero clear failures across 960 statistics over six runs and five seeds, NIST SP 800-22 (188/188), and PractRand to 4 TB without anomalies; bit-level tests also disclose a seed-specific residual bias, distinguishing battery-level success from individual-bit uniformity. The system is hyperchaotic (λ₁ = 5.7788, λ₂ = 1.0182); a Pesin-type estimate places the ideal-map entropy budget at 9.81 bits per iteration (a dynamical diagnostic, not a proven output-entropy bound). A semi-analytical scaling law λ ≈ 2 ln(K · p / 2), derived from the Gauss-Seidel Jacobian, predicts the maximal Lyapunov exponent within 0.50% across the 14 measured configurations; companion determinant laws predict the full spectrum and, within the validated parameter range, the hyperchaos boundary q > p. Patent pending: PCT/IB2026/052737, PCT/IB2026/053253, PCT/IB2026/053673, PCT/IB2026/058860.