Thermodynamics of Competing Memory Architectures: Phase Structure, Existence Theorem, and Scaling Laws

Physical systems that store binary information can do so using gradient architectures (bistable switches) or cyclic architectures (limit-cycle oscillators). These two classes have structurally different thermodynamic cost functions: gradient systems have vanishing dissipation at low noise but face Kramers-type escape costs that grow with noise amplitude δ, while cyclic systems carry a noise-independent housekeeping dissipation floor c > 0. We develop a unified framework for this architectural competition. First, we construct a thermodynamic phase diagram for a minimal bistable potential competing against a three-gene repressilator, identifying three regimes separated by structural boundaries: (I) passive barrier storage at low noise, (II) noise-assisted metastability near the Hopf bifurcation, and (III) active limit-cycle cycling at high noise. The thermodynamic crossover between regimes I and III occurs at δ* ≈ 0.91 for these models — a value 5× larger than a naive control-parameter comparison predicts, due to the repressilator's irreducible housekeeping floor. Second, we prove that a finite crossover δ* must exist for any gradient/cyclic architecture pair: the Gradient Divergence Lemma (Kramers escape forces Ẇ_G → ∞) and the Cyclic Boundedness Lemma (Floquet transverse relaxation gives Ẇ_C ≤ c + Kδ) together force a sign change in the cost difference, and the crossover follows by continuity. We confirm this for a second, topologically distinct pair (van der Pol / tilted double-well), finding δ* ≈ 0.99. Third, we derive a scaling law for the crossover location. For gradient architectures with confining potential V ~ x^{2n}, the minimum operational dissipation scales as Ẇ_G ~ δ^β with β = (3n−1)/(2n), from which the crossover follows as δ* ~ c^{1/β}. The general result, β = 1 + γ/α (where ΔV ~ a^α and ω_well ~ a^γ), defines a universality class: the exponent 1/β is universal within a potential class, the amplitude is not. The central consequence is a separation of sensitivities: δ* responds to the housekeeping floor c as a power law but to memory duration τ only logarithmically.