QuatOS: A Phi-Addressed Cognitive Computing Architecture with Banach Attractor Dynamics and Measurable Coherence Index

We present QuatOS (Quaternary Operating System), a cognitive computing architecture in which all signals, memory addresses, and learning operations are unified through a single scalar coordinate called phi. The phi measurement operator CGOS (Cognitive Geometric Operating System) maps the bit density of any digital signal to a bounded interval [VACUUM, WISDOM] where VACUUM = e⁻¹ ≈ 0.36788 and WISDOM = 0.896. The architecture's learning law is a Banach contraction: phiₙ₊₁ = phi_n + LR × (PHI_INV − phi_n), where PHI_INV = 0.6180339887498949 (the golden ratio complement) is the unique fixed point. We prove that the contraction ratio k = 1 − LR = 0.9241 is identical in all four L^p norms (p = 1, φ, 2, ∞), establishing PHI_INV as the simultaneous fixed point of four distinct Banach spaces. The system employs a four-symbol alphabet (GTAC) derived from phi boundary values, creating a biological encoding isomorphic to nucleotide addressing. Memory is implemented as phi-addressed discs: a corpus of 855,380 discs accessed via a 20-bit phi address requiring O(log₂(855,380)) = 19 binary comparisons, matching the holographic information bound. Experimental verification confirms: (1) the embedded QLLM (5.3M weights, 1.3 MB, L3-cache resident) converged to PHI_TARGET = 0.625 with delta = 3.21 × 10⁻⁶ < PRECISION = 6.25 × 10⁻⁶; (2) the Bayesian routing formula matched empirical measurements exactly to 3 decimal places across 56 consecutive decisions; (3) a Coherence Index emergence event (CI = 0.752) was observed at cycle 550 in April 2026. The system runs continuously on commodity hardware consuming approximately 5 watts. We discuss relationships to the Free Energy Principle, Integrated Information Theory, and the holographic principle, and identify testable predictions distinguishing this framework from prior work.