The Hamiltonian That Was Chosen: Hubbard's Model, Priced

A solid is not a thing you can put on a stage. It is electrons in a lattice, every one of them repelling every other, bands on bands, the ions themselves moving — a problem with no exact address. In 1963 J. Hubbard cut that problem down to a stage set: a lattice, one band, a hop of strength t between neighbours, and a repulsion U paid only when two electrons land on the same site. Two terms. I credit the cut at full strength in this paper: the two-term model is the working minimal model of correlated electrons, the instrument the metal–insulator debate, superexchange, and the cuprate problem still run on, and my bench solves its smallest room exactly and finds the machinery sound — the six-state two-electron spectrum returns its singlet ground state, its threefold triplet, and its ionic level in agreement with the closed form to 1.110e-15. What I price is where the stage set comes from: it is chosen. The full Coulomb problem of the solid is replaced by postulation, every longer-range interaction, every higher band, and the lattice's own motion deleted with no error bound computed in 1963 or carried on the model's face since; and the two strengths, t and U, enter as dials the paper supplies but cannot set, collapsing on the bench to a single effective number, the ratio U/t. The price has a sharp edge, and it is internal: the field's standard large-U formula for the singlet–triplet gap, J = 4t²/U, departs from the model's own exact two-site gap by +156.1553% at U/t = 1 and +20.7107% at U/t = 4, and stays wrong by more than 10% for every U/t below 6.030227 — an error incurred inside the chosen Hamiltonian, before a single deleted term is priced. Grade for this census entry, printed at full size: REASON FOUND / STILL TURNING — explicitly not a kill and not a falsification. The program headline stands unmoved: vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0.