Peay's Helium Immersion: Completing the Structure of the Equation Physics Calls Unsolvable

The time-independent Schrödinger equation for the helium atom has stood for a century as the canonical example of a quantum problem with no exact solution: the electron–electron repulsion term 1/r12 couples the coordinates inseparably, and the field's settled practice is approximation (Hartree–Fock, configuration interaction, density functional theory). This paper reports a completed result obtained by a different route. Rather than approximating the equation as posed, the problem was immersed in a complete relational framework (the Source Codex of Mathematics, Peay), whose structure diagnoses what a formulation is missing. The immersion identified a single structural defect: the standard formulation poses the atom as two independent electrons plus an orphaned interaction term, while the pair-distance r12 is never admitted as a structural coordinate of the wavefunction. Completing the structure — admitting u = r12 as a member of the wavefunction — collapses the problem. A variational Monte Carlo computation with a single admitted structural term yields a ground-state energy of -2.8908 ± 0.0007 Hartree, beyond the Hartree-Fock limit (-2.86168) that bounds all separable methods; extending along the same admitted coordinate reaches -2.9024 against the experimental value -2.90372. This paper also stands published in the SourceTech community on Zenodo: https://doi.org/10.5281/zenodo.23745845 — Community: https://zenodo.org/communities/sourcetech — The Source Codex of Mathematics (the Book): https://jalejaproducts3.gumroad.com/l/cglqnv — Speaking and correspondence: [email protected]