The Computation Trilemma: Thermodynamic Boundaries and Dynamical Instability of Algorithmic Assets

Abstract:This paper establishes an axiomatic theoretical framework to evaluate the dynamical stability of autonomous computational systems and derivative logical assets. Rooted in micro-scale Landauer dissipation and macro-scale second-law thermodynamic constraints, we formally state and prove the Computation Trilemma Theorem: within a compact state manifold defined by throughput speed (S), thermodynamic conversion efficiency (η), and logical consistency/stability (σ), there exists no admissible physical operator capable of driving all three coordinates simultaneously to their respective suprema. Building on this trilemma, we formulate a continuous state functional—the Recklessness Functional (ρ)—which rigorously measures the asymmetric divergence between localized marginal yield gradients and the infinite-horizon discounted expectation of systemic collapse. Non-linear dynamical analysis demonstrates that when control trajectories unilaterally maximize throughput speed, ρ inevitably breaches a non-zero critical bifurcation threshold, precipitating phase transitions into recursive logical self-cannibalization. Furthermore, extending gravitational tidal formalism to cognitive field densities, we derive the Computational Roche Limit (CRL). We establish from first principles that when an asset lacks genuine physical work density (ρ_solid → 0), its critical disruption radius diverges to infinity. Our derivations provide foundational existence criteria and Lyapunov stability bounds for the valuation of digital assets and the physical governance of hyperscale compute architectures. JEL Classification: C02, C58, D81, E42Keywords: Computation Trilemma, Recklessness Functional, Computational Roche Limit (CRL), Landauer's Principle, Non-Equilibrium Thermodynamics, Lyapunov Stability