Decidability, Observational Bridges, and Finite-Information Quantum Gravity

Abstract

Recent arguments in quantum gravity infer a need for a non-algorithmic metatheory from Godel incompleteness, while finite-information replies infer decidability from entropy bounds. We separate axiomatic completeness, dynamical reachability, observational interpretation, and physical tractability. Our main theorem-level results identify exact hyperplane reachability with a restricted unit-circle Skolem problem, prove exact decidability of finite-time expectation comparisons at effectively algebraic times by Lindemann--Weierstrass, and formulate observational bridge criteria at the correct reduction strength. In particular, syntactic independence alone entails neither empirical inertness nor empirical disagreement; a computable reduction from any undecidable set to observational truth transfers undecidability, while a reduction from true arithmetic would make observational truth non-arithmetical. As supporting dynamical results, we give a self-contained unitary specialization of established orbit-closure methods: strict approximate semialgebraic reachability is uniformly decidable for effectively algebraic finite-dimensional data, and reachable open targets have a positive asymptotic visit density. Finite dimension alone neither makes the state space finite nor ensures decidability for arbitrary computable coefficients. Entropy and state-packing results delimit what finite information can support.