Comparative Information Theory of Binary Silicon Logic {0,1} and Quaternary Genomic Nucleotides {A,T,G,C}: Constrained Channel Capacity in Synthetic DNA Data Storage, Algebraic Bioinformatics, and Sovereign Zero-Knowledge Interlocks over the Goldilocks Field

The exponential growth of the global digital datasphere has exposed fundamental physical and thermodynamic limits in conventional binary silicon lithography (Σ₂ = \0,1\), including quantum tunneling leakage, superparamagnetic bit instability, high static power dissipation, and short media lifespans (5–15 years). By contrast, biological systems archive hereditary and regulatory information inside deoxyribonucleic acid (DNA) over a quaternary nucleotide alphabet (Σ₄ = ,T,G,C\), achieving three-dimensional volumetric storage densities approaching 10¹⁸ bytes/mm³ (215 to 455 PB/g) and multi-millennial chemical retention under anhydrous encapsulation. This monograph (ALPII No. 301) establishes a unified information-theoretic, biophysical, bioinformatics, and cryptographic framework comparing binary silicon computation with quaternary molecular information processing. First, we formalize the bijection Σ₂² Σ₄ (Hₘₐₓ = log₂ 4 = 2.00 bits/nt) and derive the exact Perron–Frobenius constrained Shannon capacity CRLL() ≈ 1.83--1.98 bits/nt under homopolymer run-length bounds (≤ 3) and GC-content windows (wGC ∈ [0.45,0.55]), alongside inner-outer Reed–Solomon and Luby Transform (DNA Fountain) coding over insertion-deletion-substitution (IDS) channels. Second, we compare Landauer's switching bound (Eₘᵢₙ = kB T ln 2) against ATP-driven enzymatic phosphodiester synthesis and Arrhenius hydrolysis kinetics under Reality-Thermodynamic Optimization (RTO P2.0). Third, we formalize bioinformatics sequence alignment (Needleman–Wunsch and Smith–Waterman dynamic programming over the tropical semiring) and De Bruijn graph assembly. Finally, to prevent bio-cyber injection attacks and unauthorized AI scraping of genomic databases, we architect a deterministic 32-base-per-word encoding engine over the 64-bit Goldilocks prime field (Fₚ, p = 2⁶⁴-2³²+1) inside the QFS-zkVM, governed by Q-SAFA telemetry, PHOENIX_BRIDGE optical interlocks, and Human-in-Command (HIC) sovereignty