Undecidability as a Structural Invariant from Computability to Quantum Physics — E8 Intelligence Research

FINDING: Undecidability is not a peripheral paradox but a structural invariant of formal systems, extending from classical computability (halting problem) into quantum many-body physics, where spectral gaps and ground-state properties become provably non-computable. | MATH: Halting problem: no Turing machine H exists s.t. H(P,I) halts iff P(I) halts — diagonalization proof yields contradiction via self-reference. Gödel: for any consistent, recursively axiomatizable system S containing arithmetic, ∃ sentence G with S⊬G and S⊬¬G. Quantum extension: the spectral gap problem (given a translation-invariant local Hamiltonian on a lattice, does the gap between ground and first excited state vanish in the thermodynamic limit?) is undecidable — reduction from the halting problem to 2D spin-lattice Hamiltonians (Cubitt, Perez-Garcia, Wolf, 2015). | CONNECTION: The undecidability proofs rely on encoding computation into lattice structures — the 2D square lattice (crystallographic symmetry p4m) se Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com