The Neural Spike: Why the Action Potential Is All-or-Nothing - the Atomicity of the Fold

One axiom. One operation. Zero free parameters. A neuron’s action potential is all-or-nothing: it fires fully or not at all, and a stronger stimulus makes more spikes, not bigger ones. This paper forces the law from structure. The spike is the fold to unison at the lock threshold: the membrane state crosses the half-One, folds to the One (the full pulse), and resets. It is all-or-nothing because the fold is atomic — two-to-one, with no half-fold. You either cross the threshold and fold to unison or you do not; there is no partial fold, so no partial spike. A stronger push cannot make a bigger spike (the fold to unison is the same act), only cross threshold again sooner — raising the rate. The all-or-nothing law and rate coding both follow from the atomicity of the fold. Machine-checked; reproduces from one command. A standalone result within the Smithian Fold Theory of Everything (SFTOE). Full corpus, code, and the run-it-yourself VERIFY.md protocol: https://github.com/MettaMazza/Smithian-Fold-Theory