Physical Information Theory and Resource-Bounded Computation: Recasting Classical Undecidability Under Physical Constraints

We propose that accepting the truly physical nature of information fundamentally recasts classical undecidability problems under physical resource constraints. By recognizing that information processing is governed by physical laws—including Landauer's principle, Boltzmann entropy bounds, and thermodynamic constraints—we demonstrate that computations with finite physical resources have forecastable termination bounds under stated assumptions. This insight enables external observers to provide termination forecasts for resource-bounded computations based on consumption patterns, offering a physically-grounded approach to finite-model variants of classical problems. We extend this framework to NP-completeness, showing that exhaustive search becomes thermodynamically impossible beyond specific problem sizes. Our analysis suggests that classical undecidability results, while mathematically valid in abstract settings, lead to decidable finite-model variants when computational systems are constrained by physical resources. We demonstrate how the STEH Living Turing Machine provides a natural implementation framework for these physically-bounded computations, offering new perspectives on resource-aware computation within fundamental physical limits.