Spectral Gap Undecidability: Hamiltonian Parameters Encode Turing Machines, Proving Algorithmic Impossibility — E8 Intelligence Research

Abstract

FINDING: Spectral gap undecidability links Hamiltonian parameters to Turing machine encoding, proving that determining the energy gap of a quantum system is algorithmically impossible. | MATH: The spectral gap λ₁ - λ₀ (difference between ground and first excited energy eigenvalues) is undecidable for certain lattice Hamiltonians; reduction from halting problem via Turing machine encoding into Hamiltonian parameters (e.g., coupling constants J_ij, local fields h_i). | CONNECTION: Lattice structures (e.g., square, hexagonal, or quasicrystalline tilings) serve as the geometric substrate for encoding computational states; the undecidability arises from the infinite-dimensional Hilbert space of the lattice, mirroring the infinite tape of a Turing machine. No direct golden ratio or base-60 link, but the lattice symmetry (e.g., translational invariance broken by encoding) is critical. | DEPTH: 9 — This is a profound result: it establishes a fundamental limit on what can be known about quantum Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com