Allen-Wrench-Complete: Flat-Pack Furniture Assembly Instructions Are Turing-Complete

Flat-pack furniture ships without words. Because the manufacturer cannot enclose a translation for every buyer, the assembly sheet has to be followed by someone who shares no language, and no prior exposure to the product, with its author — the very same constraint one faces when designing a message for an alien who has never seen Earth. We argue that a notation built to be executed by such a reader is not a picture but a program, and we make the claim precise. We introduce the Flat-Pack Calculus, in which parts are an alphabet, fasteners are operations, the numbered steps are control flow, and the familiar “repeat for the remaining legs” glyph is a loop. We then prove the calculus is Turing-complete — capable, in principle, of any computation an ordinary computer can perform — by showing it faithfully simulates a cyclic tag system, the minimal machine behind the universality of Rule 110. A direct consequence follows: no procedure can decide, from the sheet and the parts alone, whether a given piece of furniture can ever be finished. Assembly is undecidable. We show that the single screw invariably left over at the end is exactly the halting witness this theory predicts, $$\#\{\text{leftover fasteners}\}=1⟺\text{the assembly has halted},$$ report a corpus study of $1,024$ manuals of which $61\%$ are Turing-complete (wardrobes disproportionately so; nightstands merely finite-state), and observe that adding a second assembler never lowers the asymptotic step count — it only raises the argument count.