Entropy, Computability, and the Observer: From the Halting Problem to Quantum Erasure

We propose an observer-based extension of computability theory, demonstrating that undecidability is not an absolute property but depends on an observer's access to the structural evolution of computation. Using a time-dependent refinement of Kolmogorov Complexity, Kt(P(x)), we define an external observer O who tracks the compression behavior of a program over time, rather than simulating its internal logic. This enables a three-valued halting classification function HO that outputs \( \)Yes, No, or Undecided based on the asymptotic behavior of Kt. This work proposes a paradigm shift in the interpretation of undecidability, reframing the Halting Problem not as an absolute computational boundary, but as a dynamic, observer-relative classification problem grounded in structural convergence. This extension is logically consistent and fully compatible with classical results~\cite{turing1936computable,chaitin1975theory}, while enabling a new class of entropy-aware, observer-based inferences. The framework aligns computability theory more closely with developments in statistical physics~\cite{zurek1989algorithmic}, quantum information~\cite{zurek2003decoherence}, and complexity theory~\cite{vitanyi2011minimum}, where inference arises not from deductive closure, but from entropy flow and observer-accessible regularity. We conclude by outlining implications for the foundations of logic, artificial intelligence, and quantum measurement, and by establishing a conceptual bridge between logical undecidability and entropy geometry as formalized in the TEQ framework~\cite{Sigtermans2025,Sigtermans2025Eigenphysics}.