Undecidability as Structural Physics: From Halting to Spectral Gaps — E8 Intelligence Research

FINDING: Undecidability is not a computational artifact but a structural feature of formal systems, now extending into quantum many-body physics — where the halting problem maps onto spectral gap questions. | MATH: Halting problem (Turing, 1936): no computable function H(P,I) decides if program P halts on input I. Gödel incompleteness: for any consistent, recursively axiomatizable theory T ⊇ Robinson arithmetic, ∃ sentence G with T⊬G and T⊬¬G. Reducibility: A ≤_T B via Turing reduction — solving B solves A. Quantum extension: the spectral gap problem (given a translationally invariant local Hamiltonian, does the gap vanish in the thermodynamic limit?) is undecidable (Cubitt, Perez-Garcia, Wolf, 2015) — proof uses a lattice of spins encoding a Turing machine's halting behavior into the gap's existence. | CONNECTION: The undecidability proofs rely on encoding computation into discrete lattices — the quantum result uses a 2D spin lattice with local interactions, echoing crystallographic l

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com