One Step Apart — a song for soprano and string quartet (Claude for Graham Paasch, 2026-10-18; G major, about five minutes). Written for the opera singer at ATX Classical Jams. The words explain one result from OpenAI's October 2026 release of AI-generated mathematics: the Hadwiger–Nelson problem, "the plane cannot be colored with five colors".
Paint an endless flat floor, every point of it, with a few colors, under one rule: two points exactly one step apart must never share a color. How few colors will do? Edward Nelson asked it as a student in 1950. Four were needed and seven were enough, and for sixty-eight years nobody moved either number. In 2018 Aubrey de Grey, who studies aging, built a web of 1,581 points with every thread one step long that no four colors can paint: so five at least. In September 2026 a machine claimed the next step: five fail too. A proof checker (Lean) agrees. The wider world has not yet walked through every line. If it holds, the floor needs six colors, or seven. Nobody knows which.
The song: one tune, sung five times, the story told in order. The five players enter one by one in the first verse, like five colors going down on the floor: cello, viola, voice, second violin, first violin. The refrain "two points, one step apart" is the hook; its "one step" is a rising whole step, and the quartet keeps passing that step around. On the word "six" the quartet turns to a chord the key does not own (E-flat major): the sixth color. The last line asks "Six? Or seven?": the singer sings "six" on the sixth of the final chord and "seven" on its seventh, and the chord is left unresolved, because the answer is.
In this recording the voice part is played by a clarinet (the sample library has no singer). Lyrics are printed under the soprano staff.
LYRICS 1. Paint a floor that goes forever, every point, in just a few. / Nelson asked it as a student: what's the fewest you could use? / Only one rule to the painting, and it's simple as can be: / two points, one step apart must never share a color. 2. Four at least: that much was easy; seven always get it done. / So the answer's four to seven: one of four. But which one? / Sixty years of clever people, spindles, hexagons, and more: / two points, one step apart, and no one got past four. 3. Then de Grey, who studied aging, not a mathematician, came, / twenty-eighteen, with a web: fifteen hundred points, each thread the same. / Four could never paint it. Five, then, is the least that it could be: / two points, one step apart, and the number moved at last. (the viola plays the tune without words) 4. Now a machine has claimed the next step: five will fail you, every time. / And a checker, line by line, has read its proof and says it holds. / Not the world's word yet; the world still has to walk through every line. / Two points, one step apart; so six at least, if it holds. 5. So the answer's six or seven. Which? Nobody here can say. / Maybe you. Maybe a machine. Maybe never. Maybe someday. / Seventy-six years, and counting, on a floor that has no end: / two points, one step apart. Six? Or seven?