Quantum verification of full-continuum Coulomb energy with unit nuclei

Myeongjin Kang

Published Oct 25, 2026• 0 citations

Abstract

We prove uniform QMA membership for the spectral-infimum decision problem for the full spinful antisymmetric Coulomb Hamiltonian with unit nuclei, distinct rational nuclear positions, unary electron number and inverse-polynomial promise gap. Nuclear repulsion is omitted. Positions and thresholds use binary rational encoding; the result assumes neither binding nor a spectral gap, geometric separation, bounded physical diameter, a selected finite basis, a fixed spin sector or an efficiently prepared low-energy state. The proof combines quantitative localization including escaped electrons, Hardy-controlled Coulomb caps, a finite kinetic cutoff, exact shifted sampling with L2 error control, and a polynomial-size verifier on arbitrary witness states. Together with the hardness theorem of the pinned source manuscript, this gives QMA completeness for the same problem. Classical sector localization, finite-basis electronic complexity, smooth-box discretization and earlier real-space simulation estimates are credited. The accompanying Lean support proves selected finite sampling, representation-average and scalar error interfaces; it does not formalize the full continuum or quantum-complexity theorem. No extension to exponentially large binary nuclear charges or worldwide priority claim is made. AI-assisted preprint by Myeongjin Kang, with no affiliation. Internal AI review is not external journal peer review or worldwide-priority certification. Identified OpenAI source inputs at revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb are stipulated true. Lean verifies scoped supporting results only, not the full theorem. Original prose: CC BY 4.0; original code and Lean: MIT; third-party rights preserved. See the supporting archive for precise assumptions, coverage, provenance and corrections.

DOI: 10.5281/zenodo.23647068

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