How few integers can force four equally spaced numbers of one colour, no matter how the chosen integers are coloured red and blue? Ronald Graham gave a striking set of twenty-seven numbers inside the interval from one to thirty-seven. This anonymous, unrefereed candidate proves that twenty-six numbers cannot do the same job inside that interval. The lower bound is encoded as a logical formula with one thousand five hundred and seventy-nine variables and five thousand seven hundred and sixty-one clauses. A checked proof certificate shows that formula is impossible to satisfy. The release does more than trust a solver: it reconstructs every arithmetic progression, counting constraint and colouring cut, checks a separate certificate for Graham's witness, validates positive examples and requires fifteen deliberate corruptions to fail. The exact conclusion is bounded: the minimum inside one to thirty-seven is twenty-seven. A smaller set of integers might still exist with a larger primitive diameter, so the unrestricted quantity W star of four and the AIM asymptotic questions remain open. The synthetic voice is a communication aid, not additional mathematical evidence.