Local Slater guides for square-lattice Fermi-Hubbard energy: a restricted PromiseBQP-completeness theorem

We explicitly carry through to the Hubbard endpoint the product-state (Hartree-Fock-type) case that Waite et al. state is available within their framework. The closest route is their Theorem 7 with a subset of cardinality one. We give explicit constants, track each guiding spinor through the reduction, and provide a finite description of the resulting site-local Slater determinant. The target is a repulsive Fermi-Hubbard Hamiltonian on a full open rectangular square grid, with spatially uniform positive hopping and repulsion, local vector fields, and the entire fixed-number sector Nf = M. With inverse-polynomial squared overlap and normalized decision precision, the resulting promise problem is PromiseBQP-complete; hardness holds on the subclass with a unique sector ground state and an inverse-polynomial spectral gap. The guiding spinors are retained from their creation-time penalty ground states and rounded to normalized dyadic-pair descriptions. The proof includes explicit global perturbation bounds and preparation and energy-estimation costs. MSC2020: Primary 68Q12; Secondary 81P68, 81V70. The archive contains the complete LaTeX source, vector figures, numerical test data, build instructions, a manifest, and SHA-256 digests. AI-use disclosure: The natural-language proofs, diagnostic code, partial formal proofs, and manuscript were generated using GPT-6 Astra and Claude Fable 5.1 under the author's direction on the research question and acceptance criteria. AI review comprised four independent audits and one heterogeneous adversarial audit across the two model families; completed human mathematical proof reviews number zero. Formal checking is peripheral: five source-side and five finite-endpoint declarations have Lean kernel proofs, Schrieffer-Wolff composition is zero of eight original targets, and one overgeneral transfer signature has a singleton counterexample. The full uniform completeness theorem is not formally proved. The author supplied the presentation revisions and remains responsible for final acceptance of the work. Appendix C distinguishes the earlier high-precision numerical example from the new certified Arb interval calculation.