Proof of an Axiom About the World When a physical axiom is proven, which recognition axioms are, and what G¨odel permits

An axiom about the world is said to be tested by its consequences and never proven. One axiom refutes this. The sentence "something is distinguished from something" is made true by any act that entertains it, so it holds in every world in which anything is asked, and no observation is involved. This gives a criterion. An axiom about the world is proven when it follows, in the background logic, from sentences of this performative kind; a structure in which those sentences hold and the axiom fails shows that the axiom is supplied. Three sentences are performatively necessary: something is distinguished, no record one asks about is complete, and a comparison has both its terms present. They build the two-state floor with its counts. The recognition axioms sort under the criterion into that forced floor and three supplied premises, each with an explicit countermodel: the magnitude reading of the floor, which yields the cost; the distinct-parts join, which yields the golden ratio; and the kept premise, which yields three dimensions. Each seam is one sentence wide. Once counts are compared as ratios, the composition law and the shape of the cost follow, so what the magnitude reading supplies is the bare comparison and one normalization. Additive composition alone forces the golden ratio in the limit; a performative argument reaches that a join has two distinct terms and stops there, because the floor's own doubling tower is a world whose joins have two distinct terms of one level, so what the join supplies is that its two parts are different levels. The kept premise is equivalent to three-dimensionality, so a reader who supplies it from experience has supplied the conclusion, and what the theorem adds is an account of what three-dimensionality is. Gödel's theorems bound what the axioms prove about themselves and touch neither the criterion nor any recognition quantity, once syntactic completeness is separated from physical completeness. A theory of everything is an axiom set proven of the world together with every physical quantity following from it. These are two tasks, and the state of each is given.