Learning an unknown bounded-degree quantum Hamiltonian from one thermal state

This preprint studies learning an unknown bounded-degree n-qubit Hamiltonian in a known Pauli basis from independent copies of a single-temperature Gibbs state. For fixed locality, interaction degree and fixed positive rational inverse temperature and coefficient bound, it gives a polynomial-resource algorithm using product single-qubit Pauli measurements, without geometry, a gap, correlation decay or a minimum nonzero coefficient. The output has rational coefficients, support inside the true support and total coefficient error at most the requested tolerance with the stated success probability. The temperature-dependent exponents are extremely large; the result is asymptotic. The refinement certificate is formalized in Lean conditionally on explicit analytic inputs. Initialization, iteration, solver, sampling and full resource bounds remain paper proofs. A companion source note preserves the separate hidden-Majorana discussion and its open calibration problem; it is not a proof of that broader learner. Research directed and curated by Matthew Liston. The mathematics, proofs, formalization and manuscript preparation substantially involve AI systems; paper-specific attribution and checking limits are disclosed in the paper and archive. No independent human proof verification, external endorsement or peer review is claimed. Licensing: original manuscript and documentation CC BY 4.0; original code MIT. The archive provides the exact file-level license mapping. Manuscript revision: 22 October 2026. Research director's note — Matthew Liston: This project is motivated by philosophical exploration and curiosity about AI-assisted research. My aim in sharing these results is to make them available for scrutiny, correction and further development, so that others can build on whatever proves useful to mathematics and science. Correspondence: atemporalwalk@gmail.com.